III. Matrices

Definition III.1 An m × n matrix is a set of numbers arranged in a rectangular array having m rows and n columns. It is written


A11 A12 · · · A1n
 A21 A22 · · · A12 
A= .
.
. 
..
 .
.
. 
.
.
.
.
Am1 Am2 · · · Amn

There are two important special cases. A 1 × n matrix (that is, a matrix with 1 row) is called a row vector.
An m × 1 matrix (that is, a matrix with 1 column) is called a column vector. Our convention will be that row indices are always written before column indices. As a memory aid, I think of matrices as being RC
(Roman Catholic or rows before columns).

§III.1. Matrix Operations
Definitions
1. Equality. For any two matrices A and B
A = B ⇐⇒ (a) A and B have the same number of rows and the same number of columns and
(b) Aij = Bij for all i, j

2. Addition. For any two m × n matrices A and B
(A + B)ij = Aij + Bij for all 1 ≤ i ≤ m, 1 ≤ j ≤ n
That is, the entry in row i, column j of the matrix A + B is defined to be the sum of the corresponding entries in A and B. Note: The sum A + B is only defined if A and B have the same number of rows and the same number of columns.
3. Scalar multiplication. For any number s and any m × n matrix A
(sA)ij = sAij for all 1 ≤ i ≤ m, 1 ≤ j ≤ n
For example
2

1
0

2
0
+
3
1

1
2×1+0
=
1
2×0+1

2×2+1
2
=
2×3+1
1

5
7

4. Matrix multiplication. For any m × p matrix A and any p × n matrix B p (AB)ik = j=1 Aij Bjk for all 1 ≤ i ≤ m, 1 ≤ k ≤ n

Note (a) AB is only defined if the number of columns of A is the same as the number of rows of B. If A is m × p and B is p × n, then AB is m × n.
(b) (AB)ik is the dot product of the ith row of A (viewed as a row vector) and the k th column of B
(c) Here is a memory aid. If you write the first factor of AB to the left of AB and the second factor above AB, then each entry (AB)ik of AB is built from the entries Aij j = 1, 2, · · · of A that are c Joel Feldman. 2011. All rights reserved.

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B is 3 × n

A is m × 3

AB is m × n

directly to its left and the entries Bjk j = 1, 2, · · · of B that are directly above it. These entries are multiplied in pairs, Aij Bjk , j = 1, 2, · · ·, starting as far from AB as possible and then the products are added up to yield (AB)ik = j Aij Bjk .
(d) If A is a square matrix, then AAA · · · A (n factors) makes sense and is denoted An .
(e) At this stage we have no idea why it is useful to define matrix multiplication in this way. We’ll get some first hints shortly.
Example III.2 Here is an matrix 
1
0 1 2 
2
3 4 5
3

example of the product of a 2 × 3 matrix with a 3 × 2 matrix, yielding a 2 × 2

3
0×1+1×2+2×3 0×3+1×2+2×1
8 4
2 =
=
3×1+4×2+5×3 3×3+4×2+5×1
26 22
1

Example III.3 Here is an example of the product of a 3 × 3 matrix with a 3 × 1 matrix, yielding a 3 × 1 matrix 
  

1 1 1 x1 x1 + x2 + x3
 1 2 3   x2  =  x1 + 2x2 + 3x3 
2 3 1 x3 2x1 + 3x2 + x3
Hence we may very compactly write the system of equations x1 + x2 + x3 = 4 x1 + 2x2 + 3x3 = 9
2x1 + 3x2 + x3 = 7 that we dealt with in Example II.2,

1
A = the matrix of coefficients  1
2

as
1
2
3

Ax = b where

 
1
x1
3  x = the column vector  x2 
1
x3

 
4
b = the column vector  9 
7

Basic Properties of Matrix Operations.
Using A, B and C to denote matrices and s and t to denote numbers
1. A + B = B + A
2. A + (B + C) = (A + B) + C
3. s(A + B) = sA + sB
4. (s + t)A = sA + tA
5. (st)A = s(tA)
6. 1A = A
7. A + 0 = A
8. A + (−1)A = 0
9. A(B + C) = AB + AC
10. (A + B)C = AC + BC

where 0 is the matrix all of whose entries are zero
(The matrix (−1)A is generally denoted −A.)

11. A(BC) = (AB)C
12. s(AB) = (sA)B = A(sB) c Joel Feldman. 2011. All rights reserved.

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These properties are all almost immediate consequences of the definitions. For example to verify property
9, it suffices to write down the definitions of the two sides of property 9
Aij (Bjk + Cjk )

Aij (B + C)jk =

[A(B + C)]ik =

j

j

[AB + AC]ik = [AB]ik + [AC]ik =

Aij Bjk + j Aij Cjk j Counterintuitive Properties of Matrix Operations that Cause Numerous Errors
13. In general AB = BA. Here are three examples. In the first AB is defined and BA is not defined. In the second AB and BA are both defined, but are not of the same size. In the third AB and BA are both defined and are of the same size, but are different.
A = [1 2]

B=

3 5
4 6

AB = [ 11 17 ]

BA is not defined

A = [1 2]

B=

3
4

AB = [ 11 ]

BA =

3
4

6
8

0 0
1 0

B=

0 1
0 0

AB =

BA =

1
0

0
0

A=

0
0

0
1

A consequence of this “unproperty” is that (A − B)(A + B) need not equal A2 − B 2 . Multiplying out
(A − B)(A + B) carefully gives AA + AB − BA − BB. The middle two terms need not cancel. For example A=
A−B =

0 0
1 0

B=

0 −1
1 0

0
0

1
0

A2 − B 2 =

A+B =

0
1

1
0

(A − B)(A + B) =

0 0
0 0
0 0
−
=
0 0
0 0
0 0
−1 0
0 1

14. In general AB may be 0 even if A and B are both nonzero. For example
A=

1
1

1
1
1
0 0
= 0, B =
= 0, AB =
= 0,
1
−1 −1
0 0

A consequence of this is that AB = AC does not force B = C even if every entry of A is nonzero. For example 1 −1
1 2
3 4
0 0
A=
, B=
=C=
and yet AB = AC =
−2 2
1 2
3 4
0 0
Example III.4 (The Form of the General Solution of Systems of Linear Equations Revisited)
We have just seen that any system of linear equations can be written
Ax = b where A is the matrix of coefficients, x is the column vector of unknowns, b is the column vector of right hand sides and Ax is the matrix product of A and x. We also saw in §II.3 that, if the ranks of [A] and [A | b] are the same, the general solution to this system is of the form x = u + c1 v1 + · · · + cn−ρ vn−ρ c Joel Feldman. 2011. All rights reserved.

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where n is the number of unknowns (that is, the number of components of x, or equivalently, the number of columns of A), ρ is the rank of A and c1 , · · · , cn−ρ are arbitrary constants. That u + c1 v1 + · · · + cn−ρ vn−ρ is a solution of Ax = b for all values of c1 , · · · , cn−ρ means that
A(u + c1 v1 + · · · + cn−ρ vn−ρ ) = b for all values of c1 , · · · , cn−ρ . By properties 9 and 12 of matrix operations, this implies that
Au + c1 Av1 + · · · + cn−ρ Avn−ρ = b
But this is true for all c1 , · · · , cn−ρ if and only if
Au = b

(Set c1 = · · · = cn−ρ = 0)

Av1 = 0 (Set c1 = 1, c2 = · · · = cn−ρ = 0 and sub in Au = b.)
.
.
.
Avn−ρ = 0 (Set c1 = · · · = cn−ρ−1 = 0, cn−ρ = 1 and sub in Au = b.)
In other words u is a solution of Ax = b and each of v1 , · · · , vn−ρ is a solution of Ax = 0.
Exercises for §III.1
1) Define
A=

1
1

2 3
2 1




−1 2
B =  −3 1 
−2 1

C = [ 2 −2 0 ]




2
D =  −11 
2

Compute all pairwise products (AA, AB, AC, AD, BA, · · ·) that are defined.

2) Compute A2 = AA and A3 = AAA for


0 a
a) A =  0 0
0 0
3) Let A =
a)
b)
c)
d)
e)

Find
Find
Find
Find
Find

1
0

 b c
0



1
b) A =  0
0


0 a
1 0
0 1

1
.
1

A2 , A3 , A4 .
Ak for all positive integers. eAt . (Part of this problem is to invent a reasonable definition of eAt .) a square root of A. That is, find a matrix B obeying B 2 = A. all square roots of A.

4) Compute Ak for k = 2, 3, 4 when



0
0
A=
0
0 c Joel Feldman. 2011. All rights reserved.

1
0
0
0

0
1
0
0


0
0

1
0

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§III.2 Matrices and Linear Transformations
Definition III.5 A transformation (a.k.a. map, a.k.a. function) T from IRn to IRm is a rule which assigns to each vector x in IRn a vector T (x) in IRm . T should be thought of as a machine: if you put a x T
T (x) vector x into the input hopper, it spits out a new vector T (x). A transformation is said to be linear if
T (sx + ty) = sT (x) + tT (y)

for all numbers s, t and vectors x, y.

Later in this section, we shall see that for each linear transformation T (x) there is a matrix MT such that
T (x) is the matrix product of MT times the column vector x. In other words, such that T (x) = MT x. First, however, we look at a number of examples, both of transformations that are not linear and transformations that are linear.
The map
T ([x1 , x2 ]) = [0, x2 ]
2
is not linear, because the two quantities
T (2x) = [0, (2x2 )2 ]
2T (x) = 2[0, (x2 )2 ] are not equal whenever x2 = 0. Another example of a map that is not linear is
Example III.6 (Translation) Define tranv (x) be the vector gotten by translating the head of the arrow x by v (while leaving the tail of the arrow fixed). In equations, tranv (x) = x + v. If translation were linear tranv (x) v x

the two expressions tranv (sx + ty) = sx + ty + v s tranv (x) + t tranv (y) = s(x + v) + t(y + v)
= sx + ty + (s + t)v would be equal for all s and t. But if v = 0 and s + t = 1, the two expressions are not equal.
We have just seen an example of a geometric operation that is not linear. Many other geometric operations are linear maps. As a result, linear maps play a big role in computer graphics. Here are some examples. Example III.7 (Projection) Define projφ (x) to be the projection of the vector x on the line in IR2 that passes through the origin at an angle φ from the x–axis. The vector ˆ = [cos φ, sin φ] is a unit vector b c Joel Feldman. 2011. All rights reserved.

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x

projφ (x)

φ

that lies on the line. So projφ (x) must have direction [cos φ, sin φ] (or its negative) and must have length x cos θ = x · ˆ where θ is the angle between x and ˆ The unique vector with the right direction and length b, b. is projφ (x) = (x · ˆ ˆ
b) b
It is easy to verify that this really is a linear transformation:

projφ (sx + ty) = (sx + ty) · ˆ ˆ b b
ˆ ˆ + t(y · ˆ ˆ
= s(x · b) b
b) b

= s projφ (x) + t projφ (y)

Writing both projφ (x) and x as column vectors cos φ x1 cos2 φ + x2 sin φ cos φ
=
sin φ x1 sin φ cos φ + x2 sin2 φ

projφ (x) = (x · ˆ ˆ = (x1 cos φ + x2 sin φ)
b) b
=

cos2 φ sin φ cos φ

sin φ cos φ sin2 φ

x1 x2 =

1 1 + cos 2φ sin 2φ sin 2φ
1 − cos 2φ
2

x1 x2 where we have used the double angle trig identities sin(2φ) = 2 sin φ cos φ cos(2φ) = cos2 φ − sin2 φ = 2 cos2 φ − 1 = 1 − 2 sin2 φ
Notice that this is the matrix product of a matrix that depends only on φ (not on x) times the column vector
x.
Example III.8 (Reflection) Define reflφ (x) to be the reflection of the vector x on the line in IR2 that passes through the origin at an angle φ from the x–axis. You can get from x to reflφ (x) by first walking

reflφ (x) projφ (x) x x

from x to projφ (x) and continuing in the same direction an equal distance on the far side of the line. In terms of vectors, to get from x to projφ (x), you have to add the vector projφ (x) − x to x. To continue an equal distance in the same direction, you have to add a second copy of projφ (x) − x. So reflφ (x) = x + 2[projφ (x) − x] = 2projφ (x) − x
We may, once again, write this as the matrix product of a matrix that depends only on φ (not on x) times the column vector x. reflφ (x) = 2
=

1 1 + cos 2φ sin 2φ sin 2φ
1 − cos 2φ
2
1 + cos 2φ sin 2φ sin 2φ
1 − cos 2φ

c Joel Feldman. 2011. All rights reserved.

x1 x2 −

−
1 0
0 1

x1 x2 = x1 x2

1 + cos 2φ sin 2φ sin 2φ
1 − cos 2φ
=

cos 2φ sin 2φ sin 2φ − cos 2φ

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x1 x2 −

1
0

0
1

x1 x2 x1 x2 Matrices

6

Example III.9 (Rotation) Define rotφ (x) to be the result of rotating the vector x by an angle φ about the origin.

rotφ (x) φ θ

x

If we denote by r the length of x and by θ the angle between x and the x–axis, then x= r cos θ r sin θ

To rotate this by φ we need only replace θ by θ + φ. rotφ (x) =
=

r cos(θ + φ) r(cos θ cos φ − sin θ sin φ) x1 cos φ − x2 sin φ
=
= r sin(θ + φ) r(sin θ cos φ + cos θ sin φ) x2 cos φ + x1 sin φ cos φ sin φ

− sin φ cos φ

x1 x2 Note that in each of examples 7, 8 and 9 there was a matrix A such that the map could be written
T (x) = Ax, where Ax is the matrix product of the matrix A times the column vector x. Every map of the form T (x) = Ax is automatically linear, because A(sx + ty) = s(Ax) + t(Ay) by properties (9) and (12) of matrix operations. We shall now show that, conversely, for each linear map T (x) there is a matrix A such that T (x) = Ax.
First we consider the case that T (x) is a linear map from IR2 (that is, two component vectors) to IR2 .
Any vector x in IR2 can be written
= x1

0
1
= x1 e1 + x2 e2
+ x2
1
0

e1 =

x1 x2 1
0

where

0
1

e2 =

Because T is linear
T (x) = T (x1 e1 + x2 e2 ) = x1 T (e1 ) + x2 T (e2 )
Define the numbers a11 , a12 , a21 , a22 by
T (e1 ) =

a11 a21 T (e2 ) =

a12 a22 Then
T (x) = x1 T (e1 ) + x2 T (e2 ) = x1

a a11 + x2 12 a22 a21

=

a11 x1 + a12 x2 a21 x1 + a22 x2

=

a11 a21 a12 a22 x1 x2 The same construction works for linear transformations from IRm to IRn . Define ei to be the column vector all of whose entries are zero, except for the ith , which is one. Note that every m component vector can be written
 
 

 

x1
0
0
1
0
1
 x2 
0
x =  .  = x1  .  + x2  .  + · · · + +xm  .  = x1 e1 + x2 e2 + · · · xm em
.
.
 . 
.
.
.
.
.
xm

0

c Joel Feldman. 2011. All rights reserved.

0

1

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For each i between 1 and m, T (ei ) is an n–component vector. Think of this vector as a column vector. Then define the matrix A = T (e1 ) T (e2 ) · · · T (em ) . That is, A is the matrix whose ith column is T (ei ). This is the matrix we want, because

 x1  x2 
Ax = T (e1 ) T (e2 ) · · · T (em )  . 
 . 
.
xm
= x1 T (e1 ) + x2 T (e2 ) + · · · + xm T (em )

(by the definition of matrix multiplication)

= T (x1 e1 + x2 e2 + · · · + xm em )
= T (x)

(by the linearity of T )

Example III.10 Let, as in Example III.9, rotφ be the linear transformation which rotates vectors in the plane by φ. From the figure

φ cos φ we see that rotφ (e1 ) =

rotφ (e1 ) sin φ e1 cos φ sin φ

rotφ (e2 )

sin φ

e2

φ cos φ

rotφ (e2 ) =

− sin φ cos φ

Hence the matrix which implements rotatiom is rotφ (e1 ) rotφ (e2 ) =

cos φ sin φ

− sin φ cos φ

This is exactly the matrix that we found in Example III.9. There is no need to memorize this matrix. This example has shown how to rederive it very quickly.
This formula T (x) = Ax is the reason we defined matrix multiplication the way we did. More generally, if S and T are two linear transformations with associated matrices MS and MT respectively (meaning that
T (x) = MT x and S(y) = MS y), then the map constructed by first applying T and then applying S obeys
S T (x) = MS T (x) = MS MT x so that the matrix associated with the composite map S T (x) is the matrix product MS MT of MS and
MT . It is traditional to use the same symbol to stand for both a linear transformation and its associated matrix. For example, the matrix associated with the linear transformation T (x) is traditionally denoted T as well, so that T (x) = T x.
Example III.11 Let, as in Example III.8, reflφ be the linear transformation which reflects vectors in the line through the origin that makes an angle φ with respect to the x–axis. Define the linear transformation
T (x) = refl π refl0 (x)
4
From the figure e2 = refl π (e1 )
4

e2

e1 = refl π (e2 )
4

e1 = refl0 (e1 ) refl0 (e2 ) c Joel Feldman. 2011. All rights reserved.

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we see that refl0 (e1 ) =

1
0

refl0 (e2 ) =

0
−1

refl π (e1 ) =
4

0
1

refl π (e2 ) =
4

1
0

so that refl0 (x) = refl0 (e1 ) refl0 (e2 ) x =

1
0

0
−1

x1 x2 refl π (y) = refl π (e1 ) refl π (e2 ) y =
4
4
4

0 1
1 0

y1 y2 The linear transformation T is thus
T (x) = refl π refl0 (x) =
4

0 1
0
refl0 (x) =
1 0
1

1
0

1
0

0
−1

x1 x2 This tells us in particular that
T (e1 ) =

0
= e2
1

=

0 −1
1 0

x1 x2 e2 = T (e1 )
T (e2 ) =

−1
= −e1
0
e1

−e1 = T (e2 )

The vectors e1 and e2 are each rotated by 90◦ . Since

refl π refl0 (ei ) = T (ei ) = rot π (ei )
4
2 for both i = 1.2, the matrix for T is the same as the matrix for rot π . So refl π refl0 (x) = rot π (x) for all
2
4
2
x. This is often written refl π ◦ refl0 = rot π where S ◦ T means “first apply T and then apply S”. That is,
4
2
S ◦ T (x) = S T (x) .
Exercises for §III.2
1) Find the matrices which project on the lines
a) x = y
b) 3x + 4y = 0
2) Find the matrices which reflect in the lines
a) x = y
b) 3x + 4y = 0
3) Find the matrices which project on the planes
a) x = y
b) x + 2y + 2z = 0
4) Find the matrices which reflect in the planes
a) x = y
b) x + 2y + 2z = 0
5) Find the matrices which rotate about the origin in IR2 by
a) π/4
b) π/2
c) π
6) Find the matrix which rotates about the z–axis in IR3 by θ.
7) The matrix

√
3+2
1 √
3−2
√
4
2

√
√3 − 2
3+2
√
2

√ 
−√2
− 2
√
12

is a rotation in IR3 . What axis does it rotate about and what is the angle of rotation?
8) Find the matrix which first reflects about the line in IR2 that makes an angle φ with the x–axis and then reflects about the line that makes an angle θ with the x–axis. Give another geometric interpretation of this matrix. c Joel Feldman. 2011. All rights reserved.

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§III.3 Another Application of Matrix Multiplication – Random Walks
“Random walks”, or more precisely “discrete random walks”, refer to a class of problems in which you are given the following information.
(H1 ) The system of interest has a finite number of possible states, labelled 1, 2, 3, · · ·, S.
(H2 ) We are interested in the system at times t = 0, 1, 2, 3, · · ·.
(H3 ) If at some time n the system is in some state j, then, at time n + 1 the system is in state i with probability pi,j . This is the case for each i = 1, 2, 3, · · · , S. The pi,j ’s are given numbers that
S
obey i=1 pi,j = 1.
(Here Hp stands for “hypothesis number p”.) Let the components of the column vector
x

n,1



 xn,2  xn =  . 
 . 
.
xn,S be the probabilities that, at time n, the system is in state 1, state 2, · · ·, state S, respectively. That is, xn,j denotes the probability that, at time n, the system is in state j. Rather than using the language of probability, you can imagine that the system consists of piles of sand located at sites 1 through S. There is a total of one ton of sand. At time n, the amount of sand at site 1 is xn,1 , the amount of sand at site 2 is xn,2 and so on. According to (H3 ), between time n and time n + 1, the fraction pi,j of the sand at site j is moved to site i. So pi,j xn,j tons of sand are moved from site j to site i by time n + 1. The total amount of sand at site i at time n + 1 is the sum, over j from 1 to S, of the amount pi,j xn,j of sand moved to site i from site j. Hence
S

pi,j xn,j

xn+1,i = j=1 In vector and matrix notation xn+1 = P xn

(III.1)

where P is the S × S matrix whose entry in row i, column j is pi,j . The pi,j ’s are called the transition probabilities for the random walk and P is called the transition matrix.
(H4 ) In a random walk problem, you are also given the initial condition x0 . Often, you will be told that at time 0, the system is in one specific state j0 . In this case, x0,j0 = 1 and x0,j = 0 for all j = j0 , so that
 
0
.
.
.
 
0
  x0 =  1  ← row j0
 
0
.
.
.
0

We now have enough information to determine the time evolution of the system. Repeatedly applying (III.1), x1 = P x0

x2 = P x1 = P 2 x0

x3 = P x2 = P 3 x0

···

xn = P n x0

···

Example III.12 (Gambler’s Ruin) I will give two descriptions of this random walk. The first description motivates the name “Gambler’s Ruin”.
Imagine that you are a gambler. At time zero you walk into a casino with a stake of $d. At that time the house has $(D − d). (So the total amount of money in play is $D.) At each time 1, 2, · · ·, you play a game of chance in which you win $1 from the house with probability w and loose $1 to the house with probability ℓ = 1 − w. This continues until either you have $0 (in which case you are broke) or c Joel Feldman. 2011. All rights reserved.

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you have $D (in which case the house is broke). That is, if at time n you have $0, then at time n + 1 you again have $0 and if at time n you have $D, then at time n + 1 you again have $D. The transition probabilities for Gambler’s Ruin are

pi,j

and the transition matrix is


1

0



w

= ℓ=1−w

0



1


0

 i=0  i=1 

i=2

 i=3 
.

P = .
.
 i = D−3

i = D−2

i = D−1 i=D 0

1

2

j=0
1
0
0
0
.
.
.
0
0
0
0

if if if if if if if

j = 0 and i = 0 j = 0 and i = 0
0 < j < D and i = j + 1
0 < j < D and i = j − 1
0 < j < D and i = j − 1, j + 1 j = D and i = D j = D and i = D

j=1 j=2
ℓ
0
0
ℓ w 0
0
w
.
.
.
.
.
.
0
0
0
0
0
0
0
0

··· j = D − 2
···
0
···
0
···
0
···
0
.
..
.
.
.
···
ℓ
···
0
··· w ···
0

ℓ

w

d

j =D−1 j =D

0
0
0
0 

0
0 

0
0 
.
. 
.
. 
.
. 
0
0 

ℓ
0 

0
0  w 1
D

The second description, of the same mathematical system, motivates why it is called a “random walk”.
At time zero, a drunk is at a location d. Once each unit of time, the drunk staggers to the right one unit with probability w and staggers to the left one unit with probability ℓ = 1 − w. This continues until the drunk reaches either the bar at 0 or the bar at D. Once the drunk reaches a bar, he remains there for ever. Here is a table giving the time evolution of Gambler’s Ruin, assuming that d = 2, D = 8, w = 0.49 and
ℓ = 0.51. (Entries are rounded to three decimal places.)
 
0
0
 
1
 
0
 
0
  x0 =  0 
 
0
 
0
 
0
 
0
0













$0
0
0.260
0.260
0.390
0.390
0.471
$1
 0.51 
 0 
 0.255 
 0 
 0.159 
 0 












$2
 0 
 0.500 
 0 
 0.312 
 0 
 0.218 












$3
 0.49 
 0 
 0.367 
 0 
 0.275 
 0 












$4
 0 
 0.240 
 0 
 0.240 
 0 
 0.210 












$5 x1 =  0  x2 =  0  x3 =  0.118  x4 =  0  x5 =  0.147  x6 =  0 












$6
 0 
 0 
 0 
 0.058 
 0 
 0.086 












$7
 0 
 0 
 0 
 0 
 0.028 
 0 












$8
 0 
 0 
 0 
 0 
 0 
 0.014 












$9
0
0
0
0
0
0
$10
0
0
0
0
0
0

Here, for example, x0 says that the gambler started with $2 at time 0. The vector x1 says that, at time 1, he has $1 with probability 0.51 and $3 with probability 0.49. The vector x2 says that, at time 2, he has $0 with probability 0.51 × 0.51 = 0.2601, $2 with probability 0.51 × 0.49 + 0.49 × 0.51 = 0.4998 and $4 with probability 0.49 × 0.49 = 0.2401. c Joel Feldman. 2011. All rights reserved.

March 7, 2011

Matrices

11

§III.4 The Transpose of a Matrix
Definition III.13 The transpose of the m × n matrix A is the n × m matrix At with matrix elements
At = Aj,i i,j for all 1 ≤ i ≤ n, 1 ≤ j ≤ m

The rows of At are the columns of A and the columns of At are the rows of A.
Example III.14 The transpose of the 2 × 3 matrix A on the left below. 
1
1 2 3
A=
At =  2
4 5 6
3

below is the 3 × 2 matrix At on the right

4
5
6

There are two important properties of the transpose operation.
1. If A is any m × n matrix, x is any m component vector and y is any n component vector x · (Ay) = (At x) · y
To see this, we just compute the left and right hand sides m m

x · (Ay) =

Ai,j yj = j=1 i=1

i=1

n

m

j=1

i=1

n

(At x) · y =

n

xi

xi (Ay)i =

(At x)j yj = j=1 xi Ai,j yj
1≤i≤m
1≤j≤n

xi At yj j,i At xi yj = j,i 1≤i≤m
1≤j≤n

and observe that they are the same, because of the definition of At .
2. If A is any ℓ × m and B is any m × n matrix, then
(AB)t = B t At
Be careful about the order of matrix multiplication here. To see this, we also just compute the left and right hand sides m (AB)t = (AB)j,i i,j =

m t t

Aj,k Bk,i k=1 m

t
Bi,k At = k,j (B A )i,j = k=1 Bk,i Aj,k k=1 and observe that they are the same.

§III.5 Matrix Inverses
Suppose A is a matrix. What is A−1 ? It’s the thing you multiply A by to get 1. What is 1?
Definition III.15 The m × m identity matrix Im (generally the subscript m is dropped from the notation) is the m × m matrix whose (i, j) matrix element is 1 if i = j and 0 if i = j.
For example
I2 = c Joel Feldman. 2011. All rights reserved.

1
0

0
1

March 7, 2011

Matrices

12

The reason we call this the identity matrix is that, for any m × n matrix A
Im A = AIn = A
It is easy to check that this is true. For example, fix any i and k with 1 ≤ i ≤ m, 1 ≤ k ≤ n. By the m definition of Iij the only nonzero term in the sum (IA)ik = j=1 Iij Ajk is that with j = i. Furthermore the
Iii that appears in that term takes the value one so (IA)ik = Iii Aik = Aik , as desired.
Definition III.16 A matrix B is called an inverse of the matrix A if AB = I and BA = I. If A has an inverse, then A is said to be invertible or nonsingular. Otherwise, it is said to be singular. The inverse is generally denoted A−1 .
Example III.17 Let’s consider the general 2 × 2 matrix a c

A=

b d and see if we can find an inverse for it. Let’s call the inverse
B=

X
Y

X′
Y′

The matrix A is to be treated as a given matrix. At this stage, B is unknown. To help keep straight what is known and what isn’t, I’ve made all of the knowns lower case and all of the unknowns upper case. To be an inverse, B must obey
AB =

a c b d X
Y

X′
Y′

=

aX + bY cX + dY

aX ′ + bY ′ cX ′ + dY ′

=

1
0

0
1

This condition consists of four equations aX + bY = 1

(1)

cX + dY = 0 aX ′ + bY ′ = 0

(2)
(1’)

cX ′ + dY ′ = 1

(2’)

in the four unknowns X, Y, X ′ , Y ′ . Note that
a) The unknowns X, Y appear only in the first two equations.
b) The unknowns X ′ , Y ′ appear only in the last two equations.
c) The coefficients on the left hand side of (1) are identical to the coefficients on the left hand side of (1’).
d) The coefficients on the left hand side of (2) are identical to the coefficients on the left hand side of (2’).
Consequently we can solve for X, Y and X ′ , Y ′ at the same time: d(1’) − b(2’) : (ad − bc)X ′ = −b c(1’) − a(2’) : (bc − ad)Y ′ = −a

d(1) − b(2) : (ad − bc)X = d c(1) − a(2) : (bc − ad)Y = c
Dividing across gives
B=

X
Y

X′
Y′

=

1 d ad − bc −c

−b a Provided ad − bc = 0, this B exists and obeys equations (1) through (2’) and hence AB = I. For B to be an inverse for A it must also obey
I = BA = c Joel Feldman. 2011. All rights reserved.

1 d ad − bc −c

−b a March 7, 2011

a c b d Matrices

13

If you go ahead and multiply out the matrices on the right, you see that this condition is indeed satisfied.
We conclude that, if det A = ad − bc = 0, the inverse of A exists and is a b c d

−1

=

1 d −b ad − bc −c a

On the other hand, if det A = ad − bc = 0, equations d(1)−b(2), d(1’)−b(2’), c(1)−a(2) and c(1’)−a(2’) force d = −b = c = −a = 0 and then the left hand side of equation (1) is zero for all values of X and Y so that equation (1) cannot be satisfied and A cannot have an inverse.
Properties of Inverses
1. If AB = I and CA = I, then B = C. Consequently A has at most one inverse.
Proof: If AB = I and CA = I, then B = IB = CAB = CI = C. If B and C are both inverses of A, then, by definition, AB = BA = I and AC = CA = I. In particular AB = I and CA = I, so that
B = C.
2. If A and B are both invertible, then so is AB and (AB)−1 = B −1 A−1 .
Proof: We have a guess for (AB)−1 . To check that the guess is correct, we merely need to check the requirements of the definition
(AB)(B −1 A−1 ) = ABB −1 A−1 = AIA−1 = AA−1 = I
(B −1 A−1 )(AB) = B −1 A−1 AB = B −1 IB = B −1 B = I
3. If A is invertible, then so is At and At

−1

t

= A−1 .

Proof: Let’s use B to denote the inverse of A (so there won’t be so many superscripts around.) By definition AB = BA = I
These three matrices are the same. So their transposes are the same. Since (AB)t = At B t , (BA)t =
At B t and I t = I, we have
B t At = At B t = I t = I which is exactly the definition of “the inverse of At is B t ”.
4. Suppose that A is invertible. Then Ax = b ⇐⇒ x = A−1 b.
Proof: Multiplying the first equation by A−1 gives the second and multiplying the second by A gives the first.
WARNING: This property is conceptually important. But it is usually computationally much more efficient to solve Ax = b by Gaussian elimination than it is to find A−1 and then multiply A−1 b.
5. Only square matrices can be invertible.
Outline of Proof: Let A be an invertible m × n matrix. Then there exists an n × m matrix B such that
AB = I, where I is the m × m identity matrix. We shall see in the next section, that the j th column of B is solves the system of equations Ax = ej , where ej is the j th column of I. Because the identity matrix I has rank m, there must exist some 1 ≤ j ≤ m such that the augmented matrix [A|ej ] also has rank m. By property 1 above, the corresponding system of equations must have a unique solution.
Consequently, the number of unknowns, n, must equal the rank, m.
6. If A is an n × n matrix then the following are equivalent:
(i) A is invertible.
(ii) For each vector b, the system of equations Ax = b has exactly one solution. c Joel Feldman. 2011. All rights reserved.

March 7, 2011

Matrices

14

(iii) x = 0 is the only solution of Ax = 0
(iv) A has rank n.
(v) There is a matrix B such that AB = I (i.e. A has a right inverse).
(vi) There is a matrix B such that BA = I (i.e. A has a left inverse).
In other words, if a given square matrix satisfies any one of properties (i)–(vi), it satisfies all of them.
Proof: The
(i)⇒(ii):
(ii)⇒(iii):
(iii)⇒(iv):
(iv)⇒(v):

notation “ (i)⇒(ii) ” means “ If statement (i) is true, then statement (ii) is also true”.
If A is invertible, then Ax = b is true if and only if x = A−1 b is true. x = 0 always solves Ax = 0. So (iii) is the special case of (ii) with b = 0.
Ax = 0 has precisely one solution ⇒ rank A = #unknowns = n .
Let A be an n × n matrix with rank n. We are to find another n × n matrix B that obeys AB = I. This is a system of linear equations for the unknown matrix B. We are more used to systems of equations with the unknowns being vectors. We can convert our
“unknown matrix” problem into “unkown vector” problems just by giving names to the columns of B and I. Of course the j th column of I is the standard unit vector ej , all of whose components are zero except for the j th , which is 1. Let’s use Bj to denote the j th column of B. We are to solve A B1 B2 · · · Bn = e1 e2 · · · en . For the two matrices
A[B1 B2 · · · Bn ] = [AB1 AB2 · · · ABn ] and [e1 e2 · · · en ] to be equal, all of their columns must agree, so the requirement AB = I may be expressed as
ABi = ei

for i = 1, · · · , n

(This argument is repeated, in more detail, with examples, in the next section.) Since A has rank n and ABi = ei is a system of n linear equations in n unknowns, it has a unique solution. (v)⇒(vi): Assume AB = I. Then Bx = 0 ⇒ ABx = 0 ⇒ x = 0. So condition (ii) is applicable to B, which implies that (iii) and subsequently (iv) are also applicable to B. This implies that there is a matrix C obeying BC = I. But C = (AB)C = A(BC) = A so BA = I.
(vi)⇒(i): Assume that BA = I. Then Ax = 0 ⇒ BAx = 0 ⇒ x = 0. So (ii) applies to A. So (iv) and (v) apply to A.
Items (ii), (iii), (iv), (v) and (vi) of property 4 are all tests for invertibility. We shall get another test, once we have generalized the definition of determinant to matrices larger than 3 × 3: a square matrix A is invertible if and only if det A = 0.
Example III.18 Let A be the matrix which implements reflection in the line y = x, let B be the matrix that implements reflection in the x axis and let C be the matrix for the linear transformation that first refelcts in the x axis and then reflects in the line y = x. We saw in Example III.11 that
A=

0 1
1 0

1 0
0 −1

B=

C = AB =

0 −1
1 0

Now the inverse of any reflection is itself. (That is, executing the same reflection twice returns every vector to its original location.) So
A−1 = A =

0
1

1
0

B −1 = B =

1
0

0
−1

(Go ahead and check for yourself that the matrix products AA and BB are both I.) So
C −1 = (AB)−1 = B −1 A−1 =

1
0

0
−1

0
1

1
0 1
=
0
−1 0

We saw in Example III.11 that C was rotation by 90◦ . So C −1 should be rotation by −90◦ . Now rotation
0 1 by −90◦ maps e1 to −e2 and e2 to e1 , which is exactly what does. −1 0 c Joel Feldman. 2011. All rights reserved.

March 7, 2011

Matrices

15

Exercises for §III.5
1) Determine which of the following matrices are invertible and, for each that is, find its inverse. Use the method of Example II.10. Do not used the canned formula derived in Example II.10.
1 1
1 3
2 4
a)
b)
c)
1 −1
−2 2
1 2
2) Either prove or find a counterexample to the statement “If BA = I and AC = I then B = C.”

§III.6 Finding Inverses
Suppose that we are given an n × n matrix A and we wish to find a matrix B obeying AB = I. Denote by Bi the (as yet unknown) ith column of B and by ei the column vector having all entries zero except for a one in the ith row. For example, when n = 2
B=

b11 b21 b12 b22 b11 b21 B1 =

B2 =

b12 b22 e1 =

1
0

e2 =

0
1

In this notation, the requirement AB = I is
A[B1 B2 · · · Bn ] = [e1 e2 · · · en ]
By the definition of matrix multiplication
A[B1 B2 · · · Bn ] = [AB1 AB2 · · · ABn ]
That is, the first column of AB is AB1 . For example, when n = 2, the first column of
AB =

a11 a21 a12 a22 b11 b21 is indeed identical to
AB1 =

b12 b22 a11 a21 a12 a22 a11 b11 + a12 b21 a21 b11 + a22 b21

=

b11 b21 =

a11 b12 + a12 b22 a21 b12 + a22 b22

a11 b11 + a12 b21 a21 b11 + a22 b21

For the two matrices A[B1 B2 · · · Bn ] = [AB1 AB2 · · · ABn ] and [e1 e2 · · · en ] to be equal, all of their columns must agree, so the requirement AB = I may be expressed as
ABi = ei

for i = 1, · · · , n

Recall that A and the ei ’s are all given matrices and that the Bi ’s are all unknown. We must solve n different systems of linear equations. The augmented matrix for system number i is [A|ei ]. We could apply Gauss reduction separately to the n systems. But because the left hand sides of all n systems are the same, we can solve the n systems simultaneously. We just form one big augmented matrix [A|e1 e2 · · · en ]. This augmented matrix is just short hand notation for the n systems of equations AB1 = e1 , AB2 = e2 , · · ·,
ABn = en . Here are two examples of this technique.
Example III.19 Let
1
1

A=

1
2

Of course we have already derived, in Example III.17, a canned formula for the inverse of a 2 × 2 matrix.
But we’ll find the inverse of A using the Gaussian elimination technique anyway, just to provide a trivial example of the mechanics of the technique. In this example, the augmented matrix is
A e1 e2 = c Joel Feldman. 2011. All rights reserved.

1 1 1
1 2 0

March 7, 2011

0
1
Matrices

16

It is important to always remember that if we were to erase all columns to the right of the vertical line except for the ith , we would have precisely the augmented matrix appropriate for the linear system ABi = ei . So any row operation applied to the big augmented matrix [A|e1 e2 ] really is a simultaneous application of the same row operation to 2 different augmented matrices [A|ei ] at the same time. Row reduce in the usual way.
The row echelon (upper triangular) form of this augment matrix is
1
0

1 1 0
1 −1 1

(2) − (1)

We could backsolve the two systems of equations separately. But it is easier to treat the two at the same time by further reducing the augmented matrix to reduced row echelon form.
1
0

0 2
1 −1

−1
1

(1) − (2)

What conclusion do we come to? Concentrate on, for example, the first column to the right of the vertical line.
In fact, mentally erase the second column to the right of the vertical line in all of the above computations.
Then the above row operations converted
A | e1 =

1
1

1 1
2 0

1 0 2
2
= I
0 1 −1
−1

to

Because row operations have no effect on the set of solutions of a linear system we can conclude that
B1 obeys

AB1 = e1

Since I B1 = B1 , we have that B1 =

if and only if it obeys

I B1 =

2
−1

2
−1
. Similarly, B1 =
. Thus
−1
1
A−1 =

2
−1

−1
1

which is exactly the matrix to the right of the vertical bar in the row reduced echelon form.
Example III.20 Now let’s compute the inverse of a matrix for which we do not already have a canned formula. Let


2 −3 2 5
 1 −1 1 2 
A=

3 2
2 1
1 1 −3 1
Form the augmented matrix



2 −3 2
 1 −1 1

3 2
2
1 1 −3

5
2
1
1

1
0
0
0

0
1
0
0

0
0
1
0


0
0

0
1

Again note that if we were to erase all columns to the right of the vertical line except for the ith , we would have precisely the augmented matrix appropriate for the linear system ABi = ei , so that any row operation applied to the big augmented matrix [A|e1 e2 · · · en ] really is a simultaneous application of the same row c Joel Feldman. 2011. All rights reserved.

March 7, 2011

Matrices

17

operation to n different augmented matrices [A|ei ] at

2 −3 2
(2) − 0.5(1)  0 0.5 0

(3) − 1.5(1) 0 6.5 −1
(4) − 0.5(1) 0 2.5 −4


2 −3 2
 0 0.5 0

(3) − 13(2) 0 0 −1
(4) − 5(2)
0 0 −4


the same time.
1
0
5
−0.5 −0.5 1
−6.5 −1.5 0
−1.5 −0.5 0

Row reduce in the usual way.

0 0
0 0

1 0
0 1

1
0
0
5
1
0
−0.5 −0.5
5
−13 1
0
2
−5 0
1


0
0

0
1


1
0
0 0
2 −3 2
5
1
0 0
 0 0.5 0 −0.5 −0.5


5
−13 1 0
0 0 −1
0
(4) − 4(3) 0 0
−18
47 −4 1
0
1
Again, rather than backsolving the four systems individually, it is easier to do all four at the same time by applying more row operations chosen to turn the left hand side into the identity matrix.


1
0
0 0
2 −3 2
5
0 0
 0 0.5 0 −0.5 −0.5 1


−(3) 0 0 1
−5 13 −1 0
0
−18 47 −4 1
0 0 0
1


2
2(2) + (4)  0

0
0

−3
1
0
0



2
0
1
0

0
0
5 1
0 −19 49 −4
0 −5 13 −1
1 −18 47 −4


0
1

0
1


1 0 0 0 22 −57 5 −1
 0 1 0 0 −19 49 −4 1 


13 −1 0
0 0 1 0 −5
0 0 0 1 −18 47 −4 1
By exactly the same argument as we used at the end of Example III.19, the inverse is the matrix to the right of the vertical bar in the row reduced echelon form. That is,


22 −57 5 −1
 −19 49 −4 1 
A−1 = 

−5
13 −1 0
−18 47 −4 1
0.5[(1) + 3(2) − 2(3) − 5(4)]

Exercises for §III.6
1) Determine which of the following matrices are invertible and, for each that is, find its inverse.




2
3 −1
1 −1 1
a)  1
2
3 
b)  −1 2 −1 
−1 −1 4
2 −1 1




1 1 1
2 1 4
c)  1 2 3 
d)  3 2 5 
1 4 9
0 −1 1
e)

g)



1 a
0 1

1
0
0


0 a
1 0
0 1

c Joel Feldman. 2011. All rights reserved.

f)
h)



a
0

0 b 1 a
0 1
0 0

where ab = 0

b c 1

March 7, 2011

Matrices

18

§III.7 Determinants – Definition
The determinant of a 1 × 1 matrix is defined by det [a11 ] = a11
The determinant of a 2 × 2 matrix is defined by det a11 a21 a12 a22 = a11 a22 − a12 a21

For n > 2, (in fact n ≥ 2) the determinant of an n × n matrix A, whose ij entry is denoted aij , is defined by n (−1)1+j a1j det M1j

det A = j=1 where M1j is the (n − 1) × (n − 1) matrix formed by deleting, from the original matrix A, the row and column containing a1j . This formula is called “expansion along the top row”. There is one term in the formula for each entry in the top row. The term is a sign times the entry times the determinant of the (n − 1) × (n − 1) matrix obtained by deleting the row and column that contains the entry. The sign alternates, starting with a +.


a11 det  a21 a31 a12 a22 a32



a13 a11 a23  = a11 det  a21 a33 a31

a12 a22 a32



a13 a11 a23  − a12 det  a21 a33 a31

= a11 (a22 a33 − a23 a32 )

a12 a22 a32



a13 a11 a23  + a13 det  a21 a33 a31

− a12 (a21 a33 − a23 a31 )

a12 a22 a32

 a13 a23  a33 + a13 (a21 a32 − a22 a31 )

Example III.21


1 det  1
3
Example

1
4
det 
2
7


2 3
0
0 2  = 1 × det
2
2 1

2
1
− 2 × det
1
3

2
1
+ 3 × det
1
3

0
2

= 1 × (0 − 4) − 2(1 − 6) + 3(2 − 0) = 12

III.22 In this example we compute, using




2 3 4
3 2 1
4
3 2 1 
 = det  3 3 3  − 2 det  2
3 3 3
8 9 12
7
8 9 12

the definition of the determinant,
2
3
9



1
4 3
3  + 3 det  2 3
12
7 8



1
4
3  − 4 det  2
12
7


3 2
3 3
8 9

As side computations, we evaluate the four 3 × 3 determinants


3 2 1
3 3
3 3
3 3 det  3 3 3  = 3 det
− 2 det
+ det
9 12
8 12
8 9
8 9 12
= 3(36 − 27) − 2(36 − 24) + (27 − 24)


4 det  2
7

=6
2 1
3 3
2 3
2 3
3 3  = 4 det
− 2 det
+ det
9 12
7 12
7 9
9 12
= 4(36 − 27) − 2(24 − 21) + (18 − 21)
= 27

c Joel Feldman. 2011. All rights reserved.

March 7, 2011

Matrices

19



4 det  2
7


3 1
3 3
2 3
2 3
3 3  = 4 det
− 3 det
+ det
8 12
7 12
7 8
8 12
= 4(36 − 24) − 3(24 − 21) + (16 − 21)



4 3 det  2 3
7 8

 = 34
2
3 3
2 3
2 3
3  = 4 det
− 3 det
+ 2 det
8 9
7 9
7 8
9
= 4(27 − 24) − 3(18 − 21) + 2(16 − 21)
= 11

So



1 2 3
4 3 2 det 
2 3 3
7 8 9
This is clearly a very tedious procedure.


4
1 
 = 6 − 2 × 27 + 3 × 34 − 4 × 11 = 10
3
12
We will develop a better one soon.

Exercises for §III.7
1) Evaluate the determinant of each of the following matrices
a)

1 3
2 4

b)

0 −3
2 −1

c)

0 1
1 0

d)

cos θ sin θ

− sin θ cos θ

2) Evaluate the determinant of each of the following matrices by expanding along the top row







2 0
0
2 1 5
7 −1 5
1 a
a)  1 −1 5 
b)  1 0 3 
c)  3 4 −5 
d)  0 1
2 3 −1
−1 2 0
2 3
0
0 0

 b c
1

§III.8 Determinants – Properties
Property E
If two rows of an n × n matrix are exchanged, the determinant is multiplied by −1.
 
 
.
.
.
.
 . 
 . 
 ai 
 aj 
 
 
 . 
 
.
det  .  = − det  . 
. 

 . 
 aj 
 ai 
 
 
.
.
.
.
.
.

where, for each 1 ≤ k ≤ n, ak is the k th row of the matrix and is an n component row vector.
Property M
Multiplying any single row of a matrix by t multiplies the determinant by t as well.


  a1 a1
 . 
 . 
.
.
 . 
 . 


  det  tai  = t det  ai 
 . 
 . 
 . 
 . 
.
. an an c Joel Feldman. 2011. All rights reserved.

March 7, 2011

Matrices

20

To multiply every entry in an n × n matrix by t, we have to apply Property M once for each row in the matrix, so we end up with a factor tn . For example det t
0

0
1 0
1 0
= t × det
= t × t × det
= t2 t 0 t
0 1

Property A
Adding any multiple of any row to any other row has no effect on the determinant.

 . 
.
.
.
.
.




 ai 
 ai + tam 

 . 

.
 = det  . 
.
det 
.

 . 





 am 
 am 
.
.
.
.
.
.


Property D
The determinant of any triangular matrix is the product of its diagonal entries.


a11
 0

det 

 0
0

∗ a22 0
0

···
···
..
.

∗
∗

· · · an−1 n−1
···
0


∗
∗ 

 = a11 a22 · · · ann

∗ 

ann

Property P det AB = det A det B
Property T det At = det A
Outline of Proof of Properties E, M, A, D
We will not give a complete proof. But we will give an outline that is sufficiently detailed that you should be able to fill in the gaps.
All of these properties follow fairly easily from another (in fact the standard) definition of determinant, which in turn follows fairly easily from our definition. For any n × n matrix A det A = σ∈Pn sgn σ A1σ(1) A2σ(2) · · · Anσ(n)

(III.2)

Here Pn is the set of all orderings of (1, 2, · · · , n). The symbol “P ” stands for “permutation”, which is the mathematical name for a reordering. So
P3 = (1, 2, 3), (1, 3, 2), (2, 1, 3), (2, 3, 1), (3, 1, 2), (3, 2, 1) c Joel Feldman. 2011. All rights reserved.

March 7, 2011

Matrices

21

We use σ(i) to denote the ith entry of the permutation σ. For example, when σ = (2, 3, 1), σ(2) = 3. The sign of a permutation, here denoted sgn σ is +1 if (1, 2, · · · , n) can be transformed into σ using an even number of exchanges of pairs of numbers. Otherwise, the sign is −1. For example (1, 2, 3) can be tranformed into (1, 3, 2) by just exchanging 2 and 3, so the sign of σ = (1, 3, 2) is −1. On the other hand, (1, 2, 3) can be tranformed into (3, 1, 2) by first exchanging 1 and 3 to yield (3, 2, 1) and then exchanging 1 and 2 to yield
(3, 1, 2). So the sign of σ = (3, 1, 2) is +1. Of course (1, 2, 3) can also be transformed into (3, 1, 2) by first exchanging 1 and 2 to yield (2, 1, 3), then exchanging 1 and 3 to yield (2, 3, 1), then exchanging 2 and 3 to yield (3, 2, 1) and finally exchanging 1 and 2 to yield (3, 1, 2). This used four exchanges, which is still even.
It is possible to prove that the number of exchanges of pairs of numbers used to transform (1, 2, 3) to (3, 1, 2) is always even. It is also possible to prove the sgn σ is well–defined for all permutations σ.
If A is a 2 × 2 matrix, the above definition is det A = A11 A22 − A12 A21 with the first term being the σ = (1, 2) contribution and the second term being the σ = (2, 1) contribution.
If A is a 3 × 3 matrix, the above definition is det A = A11 A22 A33 − A11 A23 A32 − A12 A21 A33 + A12 A23 A31 + A13 A21 A32 − A13 A22 A31 with the terms being the contributions, in order, from σ = (1, 2, 3), (1, 3, 2), (2, 1, 3), (2, 3, 1), (3, 1, 2),
(3, 2, 1).
To verify that the new and old definitions of determinant agree, it suffices to prove that they agree for 1 × 1 matrices, which is trivial, and that the new definition obeys the “expansion along the first row” formula. Outline of Proof of Property P
Once again, we will not give a complete proof. First assume that A is invertible. Then there is a sequence of row operations, that when applied to A, convert it into the identity matrix. That was the basis of the algorithm for computing A−1 that we developed in §III.6. Any row operation can be implemented by multiplication by a matrix. For example, if A is a 4 × 4 matrix, all rows of


1
0

4
0

0
1
0
0

0
0
1
0


0
0
A
0
1

with the exception of row (3) coincide with the corresponding row of A. Row (3) of the product is row (3) of A plus 4 times row (1) of A. The matrices that implement row operations are called elementary matrices.
We shall denote them by Qj here. We have
Qh · · · Q1 A = I for some sequence Q1 , · · · , Qh of row operation implementing matrices.
The next step is to check that det QC = det Q det C for any elementary matrix Q and any square matrix C. This is straight forward because we already know the effect that any row operation has on a determinant. Then
Qh · · · Q1 A = I
Qh · · · Q1 AB = B

=⇒
=⇒
=⇒

c Joel Feldman. 2011. All rights reserved.

det Qh · · · det Q1 det A = 1 det Qh · · · det Q1 det AB = det B
1
det AB = det B det A
March 7, 2011

Matrices

22

Now suppose that A is not invertible. Then there is a sequence of row operations, that when applied to
˜
A, convert it into a matrix A, whose last row is identically zero. Implementing these row operations using multiplication by elementary matrices, we have
˜˜
˜
˜
Qh · · · Q1 A = A

˜˜
˜
˜ det Qh · · · det Q1 det A = det A

=⇒

˜
Any matrix that has at least one row identically zero, like A, has determinant zero. Applying Qh · · · Q1 A = I
=⇒ det Qh · · · det Q1 det A = 1 with h = 1 and A replaced by the inverse of Q1 , we see that every elementary matrix has nonzero determinant. So we conclude that if A is not invertible it necessarily has determinant zero. Finally, observe that if A fails to be invertible, the same is true for AB (otherwise
B(AB)−1 is an inverse for A) and so both det A = 0 and det AB = 0.
Outline of Proof of Property T
By (III.2) and the definition of “transpose” n det At =

Aσ(i) i

sgn σ i=1 σ∈Pn

Concentrate on one term in this sum. By the definition of permutation, each of the integers 1, 2, · · ·, n appears exactly once in σ(1), σ(2), · · ·, σ(n). Reorder the factors in the product Aσ(1) 1 Aσ(2) 2 · · · Aσ(n) n so that the first indices, rather than the second indices are in increasing order. This can be impemented by making the change of variables i = σ −1 (j) in the product. n det At =

Aj σ−1 (j)

sgn σ j=1 σ∈Pn

If we rename the permutation σ −1 to τ and use the facts that
◦ as σ runs over all permutations exacly once, σ −1 runs over all permutations exactly once and
◦ sgn σ = sgn σ −1 we recover n det At =

Aj τ (j) = det A

sgn τ j=1 τ ∈Pn

Implications of Properties E, M, A ,D, P, T
1) If any two rows of a matrix A are the same, then det A = 0 because

  a1 a1
 . 
 . 
. 
 . 
 .
 . 
 
 b 
 b 
 . 
 .  det  .  = − det  . 
 . 
 . 
 
 
 b 
 b 
 . 
 . 
 . 
 . 
.
. an an


On the right hand side, the two rows containing b have been interchanged.
2) Thanks to Property E, a determinant may be expanded along any row. That is, for any 1 ≤ i ≤ n, n (−1)i+j aij det Mij

det A = j=1 c Joel Feldman. 2011. All rights reserved.

March 7, 2011

Matrices

23

where Mij is the (n − 1) × (n − 1) matrix formed by deleting from the original matrix A the row and column containing aij . To get the signs (−1)i+j right, you just have to remember the checkerboard


+ − + ···
− + − ···


+ − + ···
.
. . ..
.
. .
.
.
. .
Example III.23 If we expand the matrix of Example III.21 along its second row, we get


1 2 3
2 3
1 3
1 2 det  1 0 2  = −1 × det
+ 0 × det
− 2 × det
2 1
3 1
3 2
3 2 1
= −1 × (2 − 6) + 0 − 2(2 − 6) = 12

3) Thanks to Property T, a determinant may be expanded along any column too. That is, for any 1 ≤ j ≤ n, n (−1)i+j aij det Mij

det A = i=1 Example III.24 If we expand the matrix of Example III.21 along its second column, we get


1 2 3
1 2
1 3
1 3 det  1 0 2  = −2 × det
+ 0 × det
− 2 × det
3 1
3 1
1 2
3 2 1
= −2 × (1 − 6) + 0 − 2(2 − 3) = 12

3) We can now use Gaussian elimination to evaluate determinants. Properties A, E, M say that if you take a matrix U and apply to it a row operation, the resulting matrix V obeys det U = det V det U = − det V

det U =

1 k if the row operation is (i) → (i) + k(j) for some j = i if the row operation is (i) ↔ (j) for some j = i

det V

if the row operation is (i) → k(i) for some k = 0

These properties, combined with Gaussian elimination, allow us to relate the determinant of any given matrix to the determinant of a triangular matrix, that is trivially computed using Property D.
Example III.25

1 2
2 4 det 
1 3
1 3

1
0
5
3



2
1
3 A
0
= det 
6
0
9
0





2 1
2
(1)
1
0 −2 −1  (2) − 2(1) E
0
= − det 

1 4
4
(3) − (1)
0
1 2
7
(4) − (1)
0

1
A
0
= − det 
0
0
D

2
1
0
0



1
2
(1)
1
4
4 
(2)
A
0
= − det 

−2 −1
(3)
0
−2 3
(4) − (2)
0

= − 1 × 1 × (−2) × 4 = 8 c Joel Feldman. 2011. All rights reserved.

March 7, 2011


1
2
4
4 

−2 −1
2
7

2
1
0
1
2
1
0
0

(1)
(3)
(2)
(4)


1
2
(1)
4
4 
(2)

−2 −1
(3)
0
4
(4) − (3)

Matrices

24

Example III.26 Let’s redo the computation of the determinant in Example III.22, using row operations as well as the fact that we may expand along any row or column. Use Cj to denote expansion along column j and Rj to denote expansion along row j.


1
4
det 
2
7



3 4
1 2
3
2 1  A
 0 −5 −10
 = det 
3 3
0 −1 −3
9 12
0 −6 −12

1
2
M
= − 5 det  −1 −3
−6 −12

2
3
3
8

D




4
(1)
−5 −10 −15
−15  (2) − 4(1) C1
= det  −1 −3 −5 

−5 (3) − 2(1)
−6 −12 −16
−16 (4) − 7(1)



3
1 2
3
(1)
A
−5  = − 5 det  0 −1 −2  (2) + (1)
−16
0 0
2
(3) + 6(1)

= (−5){1 × (−1) × 2} = 10

4) The determinant provides a test for invertibility: Let A be an n × n matrix. Then
A is invertible

⇐⇒

det A = 0

Proof: Let R be the triangular matrix that results from the application of Gaussian elimination to the matrix A. Then, by properties E, M and A, det A = (nonzero number) det R
So
det A = 0 ⇐⇒ det R = 0
⇐⇒ the diagonal entries Rjj of R are all nonzero (Property D)
⇐⇒ rank R = n
⇐⇒ A is invertible

(See the definition of rank in §II.3)
(Property 6 of §III.5)

5) det A−1 = det A

−1

det Am = det A

m

are easy consequences of property P.
Exercises for §III.8
1) Evaluate the determinant of each of the following matrices by expanding along some row other than the first 





2 1 5
7 −1 5
1 a b
a)  1 0 3 
b)  3 4 −5 
c)  0 1 c 
−1 2 0
2 3
0
0 0 1
2) Evaluate the determinant of each of the following matrices using row reduction

a)

c)



1
1

1
1

3
 −2

1
4

−1
2
−2
1
1
−1
−3
1

c Joel Feldman. 2011. All rights reserved.


1 −1
4 8 

4 −8
1 1

2 0
5 −2 

1 1
2 −3

b)

d)

March 7, 2011



0
 −1

2
−3

1
1

1
1

1
0
−1
−2
1 1
2 4
3 9
4 16


−2 3
1 2

0 1
−1 0

1
8 

27
64
Matrices

25

3) For which values of the parameter λ does the matrix
−1 − λ
2
3
−2

5
−2

1
2

−λ

have an inverse?
4) For which values of the parameter λ does the system of linear equations
2x1 − x2 = λx1

2x1 + 5x2 = λx2 have a solution other than x1 = x2 = 0?

§III.9 Determinants – Applications
Testing for Invertibility
One of the main uses for determinants is testing for invertibility. A square matrix A is invertible if and only if det A = 0. Equivalently, (see property 4 of inverses given in §III.5), the system of equations Ax = 0 has a nonzero solution x if and only if det A = 0. This is heavily used in the eigenvalue/eigenvector problem in the next chapter.
Concise Formulae
a) If A is a square matrix with det A = 0,
A−1

= (−1)i+j

ij

det Mji det A

where Mji is the matrix gotten by deleting from A the row and column containing Aji .
WARNING: This formula is useful, for example, for studying the dependence of A−1 on matrix elements of A. But it is usually computationally much more efficient to solve AB = I by Gaussian elimination than it is to apply this formula.
Proof: The foundation for this formula is the expansion formula
(−1)i+j Aji det Mji

det A = i It is true for any j. This was implication number 2 in the last section. Let B be the matrix with
Bij = (−1)i+j det Mji
We have to show that (AB)kj is det A if j = k and 0 if j = k. But
(−1)i+j Aki det Mji

Aki Bij =

(AB)kj = i i

If k = j this is precisely the formula for det A given above. If k = j, this is the expansion along row j
˜
for the determinant of another matrix A. This other matrix is constructed from A by replacing row j
˜
˜ of A by row k of A. Row numbers j and k of A are identical so that det A = 0 by implication number
1 of Property E. c Joel Feldman. 2011. All rights reserved.

March 7, 2011

Matrices

26

Example III.27 Let



1
A = 1
2

Then


1 1
2 3
3 1

det M11 = det

2 3
1 3
1 2
= −7 det M12 = det
= −5 det M13 = det
= −1
3 1
2 1
2 3

det M21 = det

1 1
1 1
1 1
= −2 det M22 = det
= −1 det M23 = det
=1
3 1
2 1
2 3

det M31 = det

1 1
=1
2 3

det M32 = det

1 1
=2
1 3

det M33 = det

1 1
=1
1 2

so that det A = 1 × M11 − 1 × M12 + 1 × M13 = −7 + 5 − 1 = −3 and A−1



−7 2
1
1 
5 −1 −2 
=
−3
−1 −1 1

b) (Cramer’s rule) If A is a square matrix with det A = 0, the solution to Ax = b is

xi =

det A with ith column replaced by b det A

WARNING: This formula is useful, for example, for studying the dependence of A−1 b on matrix elements of A and b. But it is usually computationally much more efficient to solve Ax = b by Gaussian elimination than it is to apply this formula.
Proof: Since x = A−1 b, det A × xi = det A × A−1 b)i = det A

(−1)i+j bj det Mji

A−1 bj = ij j

j

The right hand side is the expansion along column i of the determinant of the matrix constructed from
A by replacing column number i with b.
Example III.28 Consider the system of equations of Example II.2. x1 + x2 + x3 = 4 x1 + 2x2 + 3x3 = 9
2x1 + 3x2 + x3 = 7
This system can be written in the form Ax = b with


1
A = 1
2
c Joel Feldman. 2011. All rights reserved.


1 1
2 3
3 1

 
4
b = 9
7
March 7, 2011

Matrices

27

According to Cramer’s rule

4
1
x1 = det  9
−3
7

1
1
det  1 x2 =
−3
2

1
1
x3 = det  1
−3
2

1
2
3
4
9
7
1
2
3




1
4
1 1
(1)
1
−3
=1
3  = − det  −3 −1 0  (2) − 3(1) = −
3
3
1
3
2 0 (3) − (1)



1
1 4
1
(1)
1
−3
=1
3  = − det  0 5
2  (2) − (1) = −
3
3
1
0 −1 −1 (3) − 2(1)



4
1 1 4
(1)
1
−6
=2
9  = − det  0 1 5  (2) − (1) = −
3
3
7
0 1 −1 (3) − 2(1)

c) The cross product



ˆ ı a × b =  a1 b1 Areas and Volumes


ˆ
a2 b2 
ˆ
k a3  b3 We have aready seen in §I.5 that areas of parallelograms and volumes of parallelopipeds are given by determinants. a a2 volume of parallelogram with sides a, b = det 1 b1 b2


a1 volume of parallelopiped with edges a, b, c = det  b1 c1  a3 b3  c3 a2 b2 c2

As a result, the standard change of variables formula for integrals in more than one dimension involves a determinant. §III.10 Worked Problems
Questions
1) Compute the following matrix products:
1
3

(a)


1
(d)  2
5

0
5

0
1

3
4

(b)


0
1 0
3
0 1
0

(g)

2 1
0 4

0
1

(j)

x
2

a
−1

y
3

3 5
1 2

1
2

−1 0
2 1




2 1
1 2 0 
0 1
0 −1 1
3 0

(e) [ 3

1
0

(h)



2
1 0]3
0

2 −1
0 0

b
4


3
0
5


3 3
0 
1 0
3
0 2



1 2
(c)  0 0
1 0
(f ) [ 2


5
3 2
01 0
0
2 0


3
0 4
12 1
0
8 0

3]

4
5

3
4

a c (i)

1
5


3
1
4

b d 2) Let
A=
c Joel Feldman. 2011. All rights reserved.

1
4

2 3
5 6

B=
March 7, 2011

−3 1 2
−3 2 0
Matrices

28

(a) Compute 2A, 3B, 2A + 3B and 3(2A + 3B).
(b) Compute 6A, 9B and 6A + 9B.
(c) Why are the last results in parts (a) and (b) the same?
3) Let
A=

1 2
3 −1



3
C = 1
4

2 6 1
1 −1 0

B=


2
2
1

(a) Compute AB, (AB)C, BC and A(BC) and verify that (AB)C = A(BC). So it is not necessary to bracket ABC.
(b) Can the order of the factors in the product ABC be changed?
4) Let
A=

−1 2
3 −4

B=

1
0

−1
3

(a) Compute A2 + 2AB + B 2 .
(b) Compute (A + B)2 .
(c) Account for the difference between the answers to parts a and b.
5) Define
A=

1 0
0 −1

B=

0
1

1
0

(a) Find all matrices that commute with A. That is, find all matrices M that obey AM = M A.
(b) Find all matrices that commute with B. That is, find all matrices M that obey BM = M B.
(c) Find all matrices that commute with both A and B. That is, find all matrices M that obey
AM = M A and BM = M B.
6) Suppose the matrix B obeys B 7 − 3B + I = 0. Is B invertible? If so, what is its inverse?
7) State whether each of the following statments is true or false. In each case give a brief reason.
(a) No square matrix with real entries can obey A2 = −I.
(b) The only 2 × 2 matrix that obeys A2 = 0 is A = 0.
(c) The only 2 × 2 matrices that obey A2 = A are A = 0 and A = I.
8) Suppose that a taxicab company uses the following strategy to maintain a fleet of fixed size: on December
31 of each year it sells 25% of the cars that are one year old (the lemons), 50% of all two year cars and all three year old cars; the next day, January 1, it buys enough new cars to replace those sold the previous day. (a) Let x(n) be the vector which gives the number of cars that are in each age category during year n.
Find a 3 × 3 matrix B such that x(n + 1) = Bx(n).
(b) Suppose the company starts with a fleet consisting of entirely third year cars. What proportion of the fleet will consist of third year cars in each of the following four years?
(c) Find all equilibrium fleet vectors. That is, all vectors x(n) obeying x(n + 1) = x(n).
(d) Find a matrix C that moves the fleet ahead two years. That is x(n + 2) = Cx(n).
(e) Are there any fleet vectors which repeat themselves every two years, but not every year?
9) Determine whether or not each of the following functions is a linear transformation.
(a) f (x, y, z) = 3x − 2y + 5z
(b) f (x, y, z) = −2y + 9z − 12
(c) f (x, y, z) = x2 + y 2 + z 2
(d) f (x, y, z) = [x + y, x − y, 0]
(e) f (x, y, z) = [x + y, x − y, 1]
(f) f (x, y, z) = [x, y, z 2 ]
10) Suppose that a linear transformation maps [1, 1] to [4, 7] and [1, −1] to [8, 3]. What vector does it map
[5, 14] to?
11) Is it possible for a linear transformation to map [1, 2] to [1, 0, −1], [3, 4] to [1, 2, 3] and [5, 8] to [3, 1, 6]?

c Joel Feldman. 2011. All rights reserved.

March 7, 2011

Matrices

29

12) Find the matrices representing the linear transformations
(a) f (x, y) = [x + 2y, y − x, x + y]
(b) g(x, y, z) = [x − y − z, 2y + 5z − 3x]
(c) f g(x, y, z)
(d) g f (x, y)
ˆ
13) A linear transformation maps ˆ to −ˆ,  to k and k to −ˆ. ı  ˆ ˆ ı (a) Find the matrix of the linear transformation.
(b) The linear transformation is a rotation. Find the axis of rotation.
(c) Find the angle of rotation.
(d) Show that this linear transformation really is a rotation.
14) Determine the matrices of the reflections in the following planes in IR3
(a) x + y + z = 0
(b) 2x − 2y − z = 0
15) A solid body is rotating about an axis which passes through the origin and has direction Ω = Ω1ˆ + ı ˆ The rate of rotation is Ω radians per second. Denote by x the coordinates, at some fixed
Ω2  + Ω3 k.
ˆ
time, of a point fixed to the body and by v the velocity vector of the point at that time. Find a matrix
A such that v = Ax.
16) Find the matrix of the linear transformation on IR2 which
(a) rotates each vector x clockwise by 45◦ .
(b) reflects each vector x in the x–axis and then rotates the result counterclockwise by 90◦ .
(c) reflects each vector x about the line x = y and then projects the result onto the x–axis.
17) The matrix on the left below is the inverse of the matrix on the right below. Fill in the missing entries.



4
3
2
•
4
3
1 
 5


−2 −2 −1 −1
11
6
4
3



•
−1 1
−3 1
 7

−10 5 •
−8
3 •


0
−1 

•
1

18) Find, if possible, the inverses of each of the following matrices.

(a)

1 4
2 7


1
(f )  4
7

(b)

2
4

5
8



0 0
2 3
0 0
5 6  (g) 
0 2
8 9
3 0

19) Let

(c)
1
0
0
0



1 2
(d)  2 5
1 3

3 4
4 −3


0
1

0
0


−1
3 
9



1 2
(e)  2 3
3 4


3
4
6




−1 2 p
B =  0 −1 1 
2
1 0

(a) For which values of p does B have an inverse?
(b) Find B −1 , for those values of p.

20) Suppose that, for some square matrix A, the series series notation, A0 is defined to be I.) Show that
(I − A)−1 =

c Joel Feldman. 2011. All rights reserved.

∞ n=0 ∞

An = I + A + A2 + A3 + · · · converges. (In

An

n=0

March 7, 2011

Matrices

30

21) Suppose that some square matrix obeys An = 0 for some positive integer.
(a) Find the inverse of A.
(b) Find the inverse of I − A.

22) Suppose that L is a linear transformation from IRn to IRn . Prove that its inverse, if it exists, is also linear. 

3 2 p
23) Suppose that det  0 p 1  = 10. What are the possible values of p?
1 0 2

24) Let



1
0
A=
0
3

3
4
1
∗

5
0
0
7


∗
6

2
8

where the ∗’s denote unknown entries. Find all possible values of det A.

25) Suppose that the 3 × 3 matrix A obeys det A = 5. Compute (a) det(4A) (b) det(A2 ) (c) det(4A2 )
26) Suppose that the 6 × 6 matrix A obeys A4 = 2A. Find all possible values of det A.

27) Evaluate



1
a
det  2 a a3

 a3 1 

a
2
a

a2 a3 1 a a a2 a3
1

Solutions
1) Compute the following matrix products:
1
3

(a)


1
(d)  2
5

0
5

0
1

3
4

(b)


0
1 0
3
0 1
0

(g)

2 1
0 4

0
1

(j)

x
2

a
−1

y
3

3 5
1 2

1
2



2 1
1 2 0 
0 1
0 −1 1
3 0

(e) [ 3

−1 0
2 1

1
0

b
4

(h)



2
1 0]3
0

2 −1
0 0





3 3
0 
1 0
3
0 2



5
3 2
01 0
0
2 0


3
0 4
12 1
0
8 0

(f ) [ 2


3
0
5

1 2
(c)  0 0
1 0
3]

4
5

3
4

a c (i)

1
5


3
1
4

b d Solution.
(a)
1 0
3 5
(b)
1
0



2
2 0 
0
−1 1
3

0 3
1×0+0×1 1×3+0×4
=
=
1 4
3×0+5×1 3×3+5×4

1
1×2+2×0+0×3
1 =
0×2−1×0+1×3
0

c Joel Feldman. 2011. All rights reserved.

0 3
5 29

1×1+2×1+0×0
=
0×1−1×1+1×0

March 7, 2011

Matrices

2
3

3
−1

31

(c)


1
0
1


2 3
0 4
0 12 1
0 0
8 0

 
3
1×0+2×2+3×8
1 = 0 × 0 + 0× 2+ 1 × 8
4
1×0+0×2+0×8


28 6 17
=  8 0 4 
0 4 3


1×4+2×1+3×0 1×3+2×1+3×4
0×4+0×1+1×0 0×3+0×1+1×4
1×4+0×1+0×0 1×3+0×1+0×4

(d)



1 0
2 3 1 0
0 1
5 0

(e)
[3
(f)



1×1+0×0 1×0+0×1 1×3+0×1
3 5
= 2 × 1 + 3 × 0 2 × 0 + 3 × 1 2 × 3 + 3 × 1
1 2
5×1+0×0 5×0+0×1 5×3+0×1


1 0 3 5
=  2 3 9 16 
5 0 15 25



2 3
1 0]3 0 = [3 × 2+ 1× 3 + 0 × 0
0 5
[2 3]


1×5+0×2
2 × 5 + 3 × 2
5×5+0×2

3 × 3 + 1 × 0 + 0 × 5] = [9 9]

4
= [ 2 × 4 + 3 × 5 ] = [ 23 ]
5

(g)
2
0
(h)
2
0
(i)

1
4

0
1

1
2


3
−1 0 
1
0 3
0

−1
2

0 1
1 4
=
1 0
4 8



3 5
3 2
5
0 01 0 =
0
2 0
2 0
1
5

3
4

a c b
=
d

−1 0 1
=
2 1 0

6
6



3
10 
1
0
2

7 4 1
12 8 4


2
0 =
0

a + 3c
5a + 4c

b + 3d
5b + 4d

ax − y
2a − 3

bx + 4y
2b + 12

B=

41 10
6
0

−3 1 2
−3 2 0

(j) x y
2 3

a b
=
−1 4

2) Let
A=

1
4

2 3
5 6

(a) Compute 2A, 3B, 2A + 3B and 3(2A + 3B).
(b) Compute 6A, 9B and 6A + 9B.
(c) Why are the last results in parts (a) and (b) the same?

c Joel Feldman. 2011. All rights reserved.

March 7, 2011

Matrices

32

Solution. (a)
2A =

2 4 6
8 10 12

3(2A + 3B) = 3

−9 3
−9 6

3B =

6
0

2
8

2A + 3B =

4 6
−9 3 6
−7 7 12
+
=
10 12
−9 6 0
−1 16 12

−21 21 36
−3 48 36

−7 7 12
=
−1 16 12

(b)
6A =

6 12 18
24 30 36

6A + 9B =

−27 9 18
−27 18 0

9B =

6 12 18
−27 9 18
+
=
24 30 36
−27 18 0

−21 21 36
−3 48 36

(c) Applying properties 3 and 5 of the “Basic Properties of Matrix Operations” given in §III.1, we have,
3

5

3(2A + 3B)=3(2A) + 3(3B)=(3 × 2)A + (3 × 3)B = 6A + 9B
3) Let
1 2
A=
3 −1



3
C = 1
4

2 6 1
B=
1 −1 0


2
2
1

(a) Compute AB, (AB)C, BC and A(BC) and verify that (AB)C = A(BC). So it is not necessary to bracket ABC.
(b) Can the order of the factors in the product ABC be changed?
Solution. (a)


3 2
20 17
4 4 1
4 4 1 
AB =
(AB)C =
1 2 =
46 51
5 19 3
5 19 3
4 1
BC =

16 17
2 0

1
3

A(BC) =

2
−1

16 17
=
2 0

20 17
46 51

Yup. They’re the same.
(b) The matrix product BAC is not defined because the matrix product BA is not defined owing to a matrix size mismatch.
4) Let
−1 2
3 −4

A=

B=

1
0

−1
3

(a) Compute A2 + 2AB + B 2 .
(b) Compute (A + B)2 .
(c) Account for the difference between the answers to parts a and b.
Solution. (a)
A2 =

−1 2
3 −4
−1
3

2AB = 2
B2 =
A2 + 2AB + B 2 = c Joel Feldman. 2011. All rights reserved.

1
0

−1
3

2
−4

−1
3

2
7
−10
=
−4
−15 22

1 −1
0 3
1 −1
0 3

=2
=

−1
7
−2 14
=
3 −15
6 −30

1 −4
0 9

7
−10
−2 14
1 −4
+
+
=
−15 22
6 −30
0 9
March 7, 2011

6 0
−9 1

Matrices

33

(b)
A+B =
(A + B)2 =

−1 2
1 −1
0
+
=
3 −4
0 3
3
0 1
3 −1

0 1
3 −1

1
−1

3 −1
−3 4

=

(c) (A + B)2 = A(A + B) + B(A + B) = A2 + AB + BA + B 2 , so the answer to part a minus the answer to part b ought to be
(A2 + 2AB + B 2 ) − (A2 + AB + BA + B 2 ) = AB − BA
=

−1 2
3 −4

=

−1
7
−4
6
−
=
3 −15
9 −12

1
0

−1
1
−
3
0

−1
3

−1 2
3 −4
3
1
−6 −3

This is indeed the difference.
5) Define
A=

1 0
0 −1

B=

0
1

1
0

(a) Find all matrices that commute with A. That is, find all matrices M that obey AM = M A.
(b) Find all matrices that commute with B. That is, find all matrices M that obey BM = M B.
(c) Find all matrices that commute with both A and B. That is, find all matrices M that obey
AM = M A and BM = M B.
Solution. (a) In order for both AM and M A to be defined, M must be a 2 × 2 matrix. Let a c

M=
Then
AM =
MA =

1 0
0 −1 a c

a c b d b d b a b
=
d
−c −d

1 0 a −b
=
0 −1 c −d

These are the same if and only if b = −b and c = −c which in turn is the case if and only if b = c = 0. a 0 for some numbers a and d.
So we need M to be of the form
0 d
(b) In order for both BM and M B to be defined, M must be a 2 × 2 matrix. Let
M=
Then

a c b d BM =

0
1

1
0

a c b c = d a

d b MB =

a c b d 0
1

1 b =
0
d

a c These are the same if and only if c = b and a = d. So we need M to be of the form

a b b a for some

numbers a and b. c Joel Feldman. 2011. All rights reserved.

March 7, 2011

Matrices

34

(c) In order to satisfy the conditions of both part a and part b, we need b = c = 0 and a = d, so M must be of the form a

1
0

0
1

for some number a.

6) Suppose the matrix B obeys B 7 − 3B + I = 0. Is B invertible? If so, what is its inverse?

Solution. Because B(3I − B 6 ) = 3B − B 7 = I, the matrix B is invertible with inverse 3I − B 6 .

7) State whether each of the following statments is true or false. In each case give a brief reason.
(a) No square matrix with real entries can obey A2 = −I.
(b) The only 2 × 2 matrix that obeys A2 = 0 is A = 0.
(c) The only 2 × 2 matrices that obey A2 = A are A = 0 and A = I.
Solution. (a) This statement is false . For example
A=

0 a b 0

=⇒

A2 =

ab
0

0 ab =⇒

A2 =

ab
0

0 ab =⇒

A2 =

a2
0

0 b2 So if ab = −1 (e.g. a = 1, b = −1), A2 = −I.
(b) This statement is false . For example
A=

0 a b 0

So if ab = 0 (e.g. a = 1, b = 0), A2 = 0.
(c) This statement is false . For example
A=

a
0

0 b So if a2 = a and b2 = b (e.g. a = 1, b = 0), A2 = A.
8) Suppose that a taxicab company uses the following strategy to maintain a fleet of fixed size: on December
31 of each year it sells 25% of the cars that are one year old (the lemons), 50% of all two year cars and all three year old cars; the next day, January 1, it buys enough new cars to replace those sold the previous day. (a) Let x(n) be the vector which gives the number of cars that are in each age category during year n.
Find a 3 × 3 matrix B such that x(n + 1) = Bx(n).
(b) Suppose the company starts with a fleet consisting of entirely third year cars. What proportion of the fleet will consist of third year cars in each of the following four years?
(c) Find all equilibrium fleet vectors. That is, all vectors x(n) obeying x(n + 1) = x(n).
(d) Find a matrix C that moves the fleet ahead two years. That is x(n + 2) = Cx(n).
(e) Are there any fleet vectors which repeat themselves every two years, but not every year?
Solution. (a) Pretend that it is now December 31 of year n. We have x(n)1 cars that are (almost) one year old, x(n)2 cars that are two years old and x(n)3 cars that are three years old. So, on December 31 of year n, we sell .25x1 (n) + .50x2 (n) + 1.0x3 (n) cars. The next day, January 1 of year n + 1 we buy the same number, .25x1 (n) + .50x2 (n) + 1.0x3 (n), of replacement cars. During year n + 1, these cars are all between zero and one year old. So x1 (n + 1) = .25x1 (n) + .50x2 (n) + 1.0x3 (n)
Of the x1 (n) cars that had their first birthday on January 1, .75x1 (n) remain with the company. During year n + 1, these cars are all between one and two years old. So x2 (n + 1) = .75x1 (n) c Joel Feldman. 2011. All rights reserved.

March 7, 2011

Matrices

35

Similarly, x3 (n + 1) = .50x2 (n)
In summary,
Bx(n)

1

= x1 (n + 1) = .25x1 (n) + .50x2 (n) + 1.0x3 (n)

Bx(n)

2

= x2 (n + 1) = .75x1 (n)

Bx(n)

3

= x3 (n + 1) =

So

.50x2 (n)



.25 .5
B =  .75 0
0 .5

does the job.
(b) If there are q cars in the fleet, we start with


1
0
0

 
0
x(1) =  0  q so that


.25 .5 x(2) = Bx(1) =  .75 0
0 .5


.25 .5 x(4) = Bx(3) =  .75 0
0 .5

   
1
0 q 00 = 0
0
q
0

   1 
.25 .5 1 q 4q x(3) = Bx(2) =  .75 0 0   0  =  3 q 
4
0 .5 0
0
0


 1   7 

  7   37 
1
.25 .5 1
4q
16 q
16 q
64 q
  3 q  =  3 q  x(5) = Bx(4) =  .75 0 0   3 q  =  21 q 
0
4
16
16
64
3
3
6
0
0 .5 0
0
q q 8
8
64 q

3
The proportions are 0, 0, 3 , 32 .
8
(c) In equilibrium (using I to denote the identity matrix)

x(n + 1) = x(n)
Bx(n) = Ix(n)
(B − I)x(n) = 0
If we call the components of x(n), α, β, γ respectively


   
−.75 .5
1
α
0
(B − I)x(n) =  .75 −1 0   β  =  0 
0
.5 −1 γ 0
8
4
The last equation forces β = 2γ and the second equation forces α = 3 β = 3 γ. In order to have an integer numbers of cars, γ has to be a positive multiple of 3. Let γ = 3p. The general solution is

 
8
x(n) = p  6  , p = 0, 1, 2, 3 · · ·
3

c Joel Feldman. 2011. All rights reserved.

March 7, 2011

Matrices

36

(d) We wish to find a matrix C obeying x(n+2) = Cx(n). As x(n+1) = Bx(n) and x(n+2) = Bx(n+1), we have x(n + 2) = Bx(n + 1) = B 2 x(n). So




1/4 2/4 1
1/4 2/4 1
C = B 2 =  3/4 0 0   3/4 0 0  =
0 2/4 0
0 2/4 0


7
1 
3
16
6


10 4
6 12 
0 0

(e) If Cx(n) = x(n), then


   
−9 10
4
α
0
1 
(C − I)x(n) =
3 −10 12   β  =  0 
16
6
0
−16 γ 0
The last equation forces α = 8 γ, just as in part c. The last equation minus twice the second equation
3
forces 20β − 40γ = 0 or β = 2γ just as in part c. The general solution is once again
 
8
x(n) = p  6  , p = 0, 1, 2, 3 · · ·
3

No fleet vector repeats itself every two years but not every year.
9) Determine whether or not each of the following functions is a linear transformation.
(a) f (x, y, z) = 3x − 2y + 5z
(b) f (x, y, z) = −2y + 9z − 12
(c) f (x, y, z) = x2 + y 2 + z 2
(d) f (x, y, z) = [x + y, x − y, 0]
(e) f (x, y, z) = [x + y, x − y, 1]
(f) f (x, y, z) = [x, y, z 2 ]
Solution. We use the notations x to stand for [x, y, z], x′ to stand for [x′ , y ′ , z ′ ] and f (x) to stand for f (x, y, z). Observe that sx + tx′ = [sx + tx′ , sy + ty ′ , sz + tz ′ ].
(a) f (x) = 3x − 2y + 5z is a function from IR3 to IR. For it to be linear, the two expressions f (sx + tx′ ) = 3(sx + tx′ ) − 2(sy + ty ′ ) + 5(sz + tz ′ )

sf (x) + tf (x′ ) = s(3x − 2y + 5z) + t(3x′ − 2y ′ + 5z ′ )

have to be equal for all x, x′ in IR3 and all s, t in IR. They are, so this f is linear .
(b) f (x) = −2y + 9z − 12 is a function from IR3 to IR. For it to be linear, the two expressions f (sx + tx′ ) = −2(sy + ty ′ ) + 9(sz + tz ′ ) − 12

sf (x) + tf (x′ ) = s(−2y + 9z − 12) + t(−2y ′ + 9z ′ − 12) have to be equal for all x, x′ in IR3 and all s, t in IR. They aren’t, because when x = y = z = x′ = y ′ = z ′ = 0 the first expression reduces to −12 and the second reduces to −12(s + t). These are equal only when s + t = 1. So this f is not linear .
(c) f (x) = x2 + y 2 + z 2 is a function from IR3 to IR. For it to be linear, the two expressions f (sx + tx′ ) = (sx + tx′ )2 + (sy + ty ′ )2 + (sz + tz ′ )2
2

2

2

sf (x) + tf (x′ ) = s(x2 + y 2 + z 2 ) + t(x′ + y ′ + z ′ ) have to be equal for all x, x′ in IR3 and all s, t in IR. They aren’t, because when y = z = x′ = y ′ = z ′ = 0 the first expression reduces to s2 x2 and the second reduces to sx2 . These are equal only when s2 = s or when x = 0. So this f is not linear .

c Joel Feldman. 2011. All rights reserved.

March 7, 2011

Matrices

37

(d) f (x) = [x + y, x − y, 0] is a function from IR3 to IR3 . For it to be linear, the two expressions f (sx + tx′ ) = [(sx + tx′ ) + (sy + ty ′ ), (sx + tx′ ) − (sy + ty ′ ), 0]

sf (x) + tf (x′ ) = s[x + y, x − y, 0] + t[x′ + y ′ , x′ − y ′ , 0]

= [sx + sy + tx′ + ty ′ , sx − sy + tx′ − ty ′ , 0]

have to be equal for all x, x′ in IR3 and all s, t in IR. They are, so this f is linear .
(e) f (x) = [x + y, x − y, 1] is a function from IR3 to IR3 . For it to be linear, the two expressions f (sx + tx′ ) = [(sx + tx′ ) + (sy + ty ′ ), (sx + tx′ ) − (sy + ty ′ ), 1]

sf (x) + tf (x′ ) = s[x + y, x − y, 1] + t[x′ + y ′ , x′ − y ′ , 1]

= [sx + sy + tx′ + ty ′ , sx − sy + tx′ − ty ′ , s + t]

have to be equal for all x, x′ in IR3 and all s, t in IR. The third components are equal only for s + t = 1, so this f is not linear .
(f) f (x) = [x, y, z 2 ] is a function from IR3 to IR3 . For it to be linear, the two expressions f (sx + tx′ ) = [sx + tx′ , sy + ty ′ , (sz + tz ′ )2 ]
2

sf (x) + tf (x′ ) = s[x, y, z 2 ] + t[x′ , y ′ , z ′ ]
2

= [sx + tx′ , sy + ty ′ , sz 2 + tz ′ ] have to be equal for all x, x′ in IR3 and all s, t in IR. If z ′ = 0 the third component of the first expression reduces to s2 z 2 while that of the second expression reduces to sz 2 . These agree only for z = 0 or s = 0, 1.
So this f is not linear .
10) Suppose that a linear transformation maps [1, 1] to [4, 7] and [1, −1] to [8, 3]. What vector does it map
[5, 14] to?
Solution 1. The vector
[5, 14] = 5[1, 0] + 14[0, 1] = 5 ([1, 1] + [1, −1]) +
2

14
2 ([1, 1]

− [1, −1]) =

19
2 [1, 1]

− 9 [1, −1]
2

is mapped to
19
2 [4, 7]

9
− 2 [8, 3] =

4 106
2, 2

= [2, 53]

Solution 2. Write the vectors as column, rather than row vectors. Let a c

b d be the matrix of the linear transformation. This matrix must obey a c

b d 1 a+b 4
=
=
1
c+d
7

a c b d 1 a−b 8
=
=
−1
c−d
3

This consists of a system of two equations in the two unknowns a and b as well as a system of two equations in the two unknowns c and d. Both systems are easy to solve a+b=4 c+d=7

c Joel Feldman. 2011. All rights reserved.

a−b=8

c−d=3

=⇒

a=6

=⇒

c=5 d=2

March 7, 2011

b = −2

Matrices

38

Now that we know the matrix of the linear transformation, we just have to apply it to the specified input vector. a b
5
6 −2
2
5
30 − 28
=
=
=
c d
14
5 2
53
14
25 + 28
11) Is it possible for a linear transformation to map [1, 2] to [1, 0, −1], [3, 4] to [1, 2, 3] and [5, 8] to [3, 1, 6]?
Solution. [5, 8] = 2[1, 2] + [3, 4]. So for the map to be linear, it is necessary that [3, 1, 6] = 2[1, 0, −1] +
[1, 2, 3] = [3, 2, 1], which is false. So, it is not possible .
12) Find the matrices representing the linear transformations
(a) f (x, y) = [x + 2y, y − x, x + y]
(b) g(x, y, z) = [x − y − z, 2y + 5z − 3x]
(c) f g(x, y, z)
(d) g f (x, y)
Solution. Writing column, rather than row vectors.
(a)

 



x + 2y a b ax + by
 y − x  =  c d  x =  cx + dy  y x+y e f ex + f y provided we choose a = 1, b = 2, c = −1, d = 1, e = 1, f = 1. So the matrix is



1 2
 −1 1 
1 1
(b)
x−y−z a =
2y + 5z − 3x d   x b c   ax + by + cz y = e f dx + ey + f z z provided we choose a = 1, b = −1, c = −1, d = −3, e = 2, f = 5. So the matrix is
1 −1 −1
−3 2
5
(c) Substituting

into

 
  x x u 1 −1 −1  
= g  y  = y v
−3 2
5
z z f

u v gives




1 2 u =  −1 1  v 1 1


   
    

 
−5 3 x 1 2 x 1 2 x 1 −1 −1   f g  y  =  −1 1  g  y  =  −1 1  y =  −4 3
−3 2
5
−2 1 z 1 1 z 1 1 z (d) Substituting

c Joel Feldman. 2011. All rights reserved.

 u v=f w 

x y   
9
x
6 y 
4
z


1 2 x =  −1 1  y 1 1


March 7, 2011

Matrices

39

into




  u u
1 −1 −1   g  v  = v −3 2
5
w w gives

g f

x y 1 −1 −1
=
f
−3 2
5



1 2
1 −1 −1  x =
−1 1 
=
−3 2
5
y
1 1

x y 1 0
0 1

x y ˆ
13) A linear transformation maps ˆ to −ˆ,  to k and k to −ˆ. ı  ˆ ˆ ı (a) Find the matrix of the linear transformation.
(b) The linear transformation is a rotation. Find the axis of rotation.
(c) Find the angle of rotation.
(d) Show that this linear transformation really is a rotation.
Solution. (a) The matrix must obey


a
d
g

  

b c
1
0 e f   0  =  −1  h i
0
0



a
d
g

The desired matrix is

b e h

    c 0
0
f 1 = 0 i 0
1



a
d
g

b e h

  

c
0
−1 f 0 =  0  i 1
0




0 0 −1
T =  −1 0 0 
0 1 0

(b) Perform the following little experiment. Take a book. Rotate the book, about its spine, by 45◦ .
Observe that the spine of the book does not move at all. Vectors lying on the axis of rotation, do not change when the rotation is executed. To find the axis of rotation of T , we just need to find a nonzero vector n obeying T n = n.


   
0 0 −1 n1 n1
 −1 0 0   n2  =  n2 
0 1 0 n3 n3

=⇒

  
−n3
n1
 −n1  =  n2  n2 n3


=⇒



 n1 1
 n2  = c  −1  for any c n3 −1


(c) Rotate the book, about its spine, by 45◦ again. Observe that the bottom and top edges of the book rotate by 45◦ . Under a rotation of θ◦ , vectors perpendicular to the axis of rotation rotate by θ◦ . Vectors that are neither perpendicular to nor parallel to the axis of rotation, rotate by angles that are strictly between 0◦ and θ◦ . (Repeat the book experiment a few times, concentrating on vectors that are almost parallel to the spine and then on vectors that are almost perpendicular to the spine, to convince yourself that this is true.) So to determine the angle of rotation, we select a vector, v, perpendicular to the axis of rotation, [1, −1, −1], and compute the angle between v and T v. The vector v = ˆ +  is perpendicular ı ˆ to the axis of rotation, because [1, 1, 0] · [1, −1, −1] = 0. It gets mapped to


  

0 0 −1
1
0
 −1 0 0   1  =  −1 
0 1 0
0
1

The angle, θ, between ˆ +  and the vector −ˆ + k that it is mapped to obeys ı ˆ
 ˆ
(ˆ + ) · (−ˆ + k) = ˆ +  ı ˆ
 ˆ ı ˆ

c Joel Feldman. 2011. All rights reserved.

−  + k cos θ
ˆ ˆ

=⇒

March 7, 2011

2 cos θ = −1

=⇒

Matrices

θ = 120◦

40

Remark. We have already seen, in part b, that every vector parallel to ˆ −  − k gets mapped to itself. ı ˆ ˆ
That is, the linear transformation does not move it at all. If the linear transformation really is a rotation then every vector perpendicular to ˆ−  − k, i.e. every vector cˆ+aˆ+bk obeying (cˆ+aˆ+bk)·(ˆ−  − k) = ı ˆ ˆ ı  ˆ ı  ˆ ı ˆ ˆ
ˆ
c − a − b = 0, i.e. every vector of the form (a + b)ˆ + aˆ + bk, should get mapped to a vector which is ı  perpendicular to ˆ−  − k, has the same length as (a + b)ˆ+ aˆ+ bk and makes an angle 120◦ with respect ı ˆ ˆ ı  ˆ
ˆ
ˆ to (a + b)ˆ + aˆ + bk. The vector (a + b)ˆ + aˆ + bk gets mapped to ı  ı 


 

0 0 −1 a+b −b
 −1 0 0   a  =  −a − b 
0 1 0 b a
This vector is again perpendicular to ˆ−  − k, by the dot product test. The lengths (a + b)ˆ+ aˆ+ bk = ı ˆ ˆ ı  ˆ
2 + a2 + b2 and − bˆ − (a + b)ˆ+ ak =
2 + (a + b)2 + a2 both equal 2(a2 + b2 + ab). The
ˆ
(a + b) b ı

dot product
ˆ
ˆ
[(a + b)ˆ + aˆ + bk] · [−bˆ − (a + b)ˆ + ak] = −b(a + b) − a(a + b) + ab ı  ı 
= −a2 − b2 − ab

ˆ
= − 1 (a + b)ˆ + aˆ + bk ı 
2

ˆ
− bˆ − (a + b)ˆ + ak ı 

shows that the angle between the two vectors is 120◦ .
14) Determine the matrices of the reflections in the following planes in IR3
(a) x + y + z = 0
(b) 2x − 2y − z = 0
ˆ is perpendicular to the given plane. The projection of any vector
Solution. (a) The vector n = ˆ +  + k ı ˆ x on n is
 
1
n·x x+y+z   projn (x) = n= 1 n 2
3
1
The reflection is

refl(x) projn (x) x  
 

 x 2 x − 2y − 2z x+y+z   1 x − 2projn (x) =  y  −
2 = y − 2x − 2z 
3
3 z 2 z − 2x − 2y

  
1 −2 −2 x 1
=
−2 1 −2   y 
3
−2 −2 1 z Check: The vector n, which is perpendicular to the plane, should be mapped to −n. On the other hand, the vectors ˆ −  and  − k, both of which are parallel to the plane, should be mapped to themselves. ı ˆ
ˆ ˆ

 
 
1 −2 −2
1
1
1
−2 1 −2   1  = −  1 
3
−2 −2 1
1
1


 

1 −2 −2
1
1
1
−2 1 −2   −1  =  −1 
3
−2 −2 1
0
0


 

1 −2 −2
0
0
1
−2 1 −2   1  =  1 
3
−2 −2 1
−1
−1 c Joel Feldman. 2011. All rights reserved.

March 7, 2011

Matrices

41

(b) The vector n = 2ˆ − 2ˆ − k is perpendicular to the given plane. The projection of any vector x on n ı  ˆ is 

2
2x − 2y − z  n·x n=
−2  projn (x) = n 2
9
−1
The reflection is

 





x
2
x + 8y + 4z
1
1
2x − 2y − z 
1
x − 2projn (x) =  y  − 2
−2  = y + 8x − 4z  =
8
9
9
9 z −1
7z + 4x − 4y
4

   x 8
4
1 −4   y  z −4 7

Check: The vector n, which is perpendicular to the plane, should be mapped to −n. On the other hand,
ˆ
the vectors ˆ +  and ˆ + 2k, both of which are parallel to the plane, should be mapped to themselves. ı ˆ ı 





1 8
4
2
2
1
8 1 −4   −2  = −  −2 
9
4 −4 7
−1
−1

   
1 8
4
1
1
1
8 1 −4   1  =  1 
9
4 −4 7
0
0

   
1 8
4
1
1
1
8 1 −4   0  =  0 
9
4 −4 7
2
2
15) A solid body is rotating about an axis which passes through the origin and has direction Ω = Ω1ˆ + ı ˆ
Ω2  + Ω3 k. The rate of rotation is Ω radians per second. Denote by x the coordinates, at some fixed
ˆ
time, of a point fixed to the body and by v the velocity vector of the point at that time. Find a matrix
A such that v = Ax.
Solution. We saw in §I.7 that the velocity vector is


ˆ ı v = Ω × x =  Ω1 x 
ˆ
Ω2 y 
 

ˆ
0
Ω2 z − Ω3 y k Ω3  =  −Ω1 z + Ω3 x  =  Ω3
−Ω2
Ω1 y − Ω2 x z −Ω3
0
Ω1

  
Ω2
x
−Ω1   y 
0
z

16) Find the matrix of the linear transformation on IR2 which
(a) rotates each vector x clockwise by 45◦ .
(b) reflects each vector x in the x–axis and then rotates the result counterclockwise by 90◦ .
(c) reflects each vector x about the line x = y and then projects the result onto the x–axis.
Solution.
1
1
ı ˆ
ˆ
ı ˆ
(a) The vector ˆ should be mapped to √2 (ˆ − ) and the vector  should be mapped to √2 (ˆ + ). The ı matrix
1 1
1
√
2 −1 1 does the job.
(b) The vector ˆ is reflected to ˆ and then rotated to  and the vector  is reflected to −ˆ and then rotated ı ı
ˆ
ˆ

to ˆ. So, in the end, ˆ is mapped to  and  is mapped to ˆ. The matrix ı ı
ˆ
ˆ ı 0
1
c Joel Feldman. 2011. All rights reserved.

1
0

March 7, 2011

Matrices

42

does the job.
(c) The vector ˆ is reflected to  and then projected to 0 and the vector  is reflected to ˆ and then ı ˆ
ˆ
ı projected to ˆ. So, in the end, ˆ is mapped to 0 and  is mapped to ˆ. The matrix ı ı
ˆ
ı
0
0

1
0

does the job.
17) The matrix on the left below is the inverse of the matrix on the right below. Fill in the missing entries.



4
3
2
•
4
3
1 
 5


−2 −2 −1 −1
11
6
4
3



•
−1 1
−3 1
 7

−10 5 •
−8
3 •


0
−1 

•
1

Solution. The matrices must obey



4
3
2
a b −1 1
4
3
1  7
−3 1
 5


−2 −2 −1 −1
−10 5 c
11
6
4
3
−8
3 e

 
0
1
−1   0
=
d
0
1
0

0
1
0
0

0
0
1
0


0
0

0
1

In particular, using (m, n) to denote the equation that matches the matrix element in row m and column n of the product of the two matrices on the left with the corresponding matrix element of the identity matrix on the right
(3, 4) :
(1, 4) :

(−2)(0) + (−2)(−1) +
(4)(0) + (3)(−1) +

(3, 1) :
(2, 3) :

(−2)(b) +
(5)(1) +

(−1)(1) = 0
(a)(1) = 0

=⇒
=⇒

(−2)(7) + (−1)(−10) + (−1)(−8) = 0
(4)(1) +
(3)(c) +
(1)(e) = 0

=⇒
=⇒

(−2)(1) +

(3, 3) : (−2)(1) +

(−1)d +
(2)d +

=⇒

(−1)(c) + (−1)(e) = 1

d=1 a=1 b=2
3c + e = −9 c + e = −5

The solution is a = 1, b = 2, c = −2, d = 1, e = −3 .
18) Find, if possible, the inverses of each of the following matrices.
1 4
2 7

(a)



1
(f )  4
7

(b)

2
4

5
8



0 0
2 3
0 0
5 6  (g) 
0 2
8 9
3 0

(c)
1
0
0
0



1 2
(d)  2 5
1 3

3 4
4 −3


0
1

0
0


−1
3 
9



1 2
(e)  2 3
3 4


3
4
6

Solution. We have a canned formula for the inverse of 2 × 2 matrices. But I’ll use row reduction, just for practice.
(a)
1
2

4 1 0
7 0 1

(1)
(2) − 2(1)

Check:

c Joel Feldman. 2011. All rights reserved.

1 4
1
0 −1 −2

1 4
2 7

0
1

(1) + 4(2)
−(2)

−7 4
1
=
2 −1
0
March 7, 2011

1 0
0 1

−7 4
2 −1

0
1
Matrices

43

(b)
2 5 1
4 8 0

0
1

(1)
(2) − 2(1)

Check:

2
0

5
1 0
−2 −2 1

(1)/2 + 5(2)/4
−(2)/2

2
4

5
8

1
0

4/3
1/3 0
−25/3 −4/3 1

−2
1

1
0

−2 5/4
1 −1/2

0
1

5/4
1 0
=
−1/2
0 1

(c)
3
4

4 1
−3 0

0
1

(1)/3
(2) − 4(1)/3

Check:

3
4

4
−3

(1) + 4(2)/25
−3(2)/25

1
0

0
1

3/25 4/25
4/25 −3/25

3/25 4/25
1 0
=
4/25 −3/25
0 1

(d)


1 2
2 5
1 3

−1 1
3 0
9 0


(1) + (3)/5 1
(2) − (3)  0
0
(3)/5
Check:

1 2
2 3
3 4

1
2
1
3 1
4 0
6 0

0
1
0


1
(1) − 3(3)
−(2) − 2(3)  0
0
(3)

Check:

(f)


1
4
7

2 3 1 0
5 6 0 1
8 9 0 0



2 −1 1 0 0
(1)
1 2
0 1
1 5 −2 1 0 
(2)
1 10 −1 0 1 (3) − (2) 0 0


2 0 6/5 −1/5 1/5
2
−1 
1 0 −3
0 1 1/5 −1/5 1/5


(e)




0 0
(1)
1
1 0  (2) − 2(1)  0
0 1
(3) − (1)
0


(1) − 2(2) 1

(2)
0
(3)
0

0 0
1 0
0 1


 
2 −1
36/5 −21/5 11/5
1
5 3   −15/5 10/5 −5/5  =  0
3 9
1/5
−1/5
1/5
0


36/5 −21/5 11/5

−3
2
−1 
1/5 −1/5 1/5

0 0
1 0
0 1





0
(1)
1 0 0
1 2
3
(1)
1
0
0  (2) − 2(1)  0 −1 −2 −2 1 0 
(2)
1 (3) − 3(1) 0 −2 −3 −3 0 1 (3) − 2(2) 0


2 0 −2 6 −3
3 −2 
1 0 0
0 1 1 −2 1


1 2
2 3
3 4


0
(1)
1
0  (2) − 4(1)  0
1 (3) − 7(1) 0

c Joel Feldman. 2011. All rights reserved.


(1) − 2(2) 1

(2)
0
(3)
0


3
−2
4 0
6
1

0 0
1 0
0 1

1
2
3
−3 −6 −4
−6 −12 −7


0
0
1


0
0
1



0 0
(1)
1
0
1 0
(2)
0 1 (3) − 2(2) 0

March 7, 2011

1
0
2
3
−1 −2 −2 1
1 −2
0
1


−2 0
1

0
3 −2 
1 −2 1

 
0
1
1 0
3 −2  =  0 1
−2 1
0 0


0 0
−1 1
5 −2 1 0 
1 −1 1
5


1
0 0
2
3
−3 −6 −4 1 0 
1 −2 1
0
0

Matrices

44

The last equation has a zero left hand side and nonzero right hand side and so cannot be satisfied. There is no inverse . As a check, observe that



  
2 3
1
0
5 6   −2  =  0 
8 9
1
0

1
4
7
No invertible matrix can map
(g)

0 0 1 0
0 0 0 1

0 2 0 0
3 0 0 0
Check:



0
0

0
3

a nonzero vector to the zero vector.
1
0
0
0
0
0
2
0

0
1
0
0


0
0
10

0
1
0
0

1
0
0
0


(4)/3 1 0
(3)/2  0 1

(1)  0 0
(2)
0 0


0
0

0
1

0
0
1
0

19) Let

0
0
1
0

0
0
0
1

 
0 0 1/3
1
0 1/2 0   0
=
0 0
0
0
1 0
0
0

0
0
1
0
0
1
0
0


0 0 1/3
0 1/2 0 

0 0
0 
1 0
0
0
0
1
0


0
0

0
1



(a) For which values of p does B have
(b) Find B −1 , for those values of p.
Solution.

−1 2 p
 0 −1 1
2
1 0


−1 2 p
B =  0 −1 1 
2
1 0

an inverse?


0 0
1 0
0 1

1
0
0


(1)
−1 2 p 1 0
 0 −1
(2)
1
0 1
(3) + 5(2)
0
0 2p + 5 2 5


(1)
−1
 0
(2)
(3) + 2(1)
0

2 p 1
−1 1 0
5 2p 2


0
0
1


0 0
1 0
0 1

There is an inverse if and only if 2p + 5 = 0 . In this case, we can continue

−(1)
1 −2 −p
 0 1 −1
−(2)
(3)/(2p + 5) 0 0
1

(1) + 2(2) 1 0

(2)
0 1
(3)
0 0

0
0
1

−1
0
2
2p+5

−1
2p+5
2
2p+5
2
2p+5

0
−1

5
2p+5

p
2p+5
−2p
2p+5
5
2p+5


(1) + p(3) 1 −2

(2) + (3)  0 1
0 0
(3)


0
0 

1
2p+5

p+2
2p+5
1
2p+5
1
2p+5



−5
2p+5
2
2p+5
2
2p+5

5p
2p+5
−2p
2p+5
5
2p+5

p
2p+5
1
2p+5
1
2p+5







∞ n=0 20) Suppose that, for some square matrix A, the series series notation, A0 is defined to be I.) Show that
−1

(I − A) c Joel Feldman. 2011. All rights reserved.

0
0
1

=

∞

An = I + A + A2 + A3 + · · · converges. (In

An

n=0

March 7, 2011

Matrices

45

Solution. We have to verify that
(I − A)

∞

∞

An =

n=0

n=0

An (I − A) = I

For the left hand side
(I − A)

∞

An =

n=0

n=0

The second sum

∞ n=0 (I − A)
∞
n=0

An − A

∞

∞

An =

n=0

n=0

An+1 = A + A2 + A3 + · · · =

so

The argument for

∞

∞

An =

∞ n=0 n=0

An −

∞

An −

∞

∞

An+1

n=0

An

n=1

An = I

n=1

An (I − A) is similar.

21) Suppose that some square matrix obeys An = 0 for some positive integer.
(a) Find the inverse of A.
(b) Find the inverse of I − A.

Solution. (a) Trick question!! A has no inverse. If A had an inverse then multiplying both sides of
An = 0 by (A−1 )n would give A−1 · · · A−1 A · · · A = 0 (with n A−1 ’s and n A’s) and then I = 0.
(b) From problem 20, we would guess (I − A)−1 = I + A + A2 + · · · + An−1 . All the remaining terms in the series I + A + A2 + · · · vanish because of An = 0. To verify that the guess is correct we multiply out (I − A)(I + A + A2 + · · · + An−1 ) = (I + A + A2 + · · · + An−1 ) − A(I + A + A2 + · · · + An−1 )
= (I + A + A2 + · · · + An−1 ) − (A + A2 + A3 + · · · + An )
= I − An = I

The same argument also gives (I + A + A2 + · · · + An−1 )(I − A) = I

22) Suppose that L is a linear transformation from IRn to IRn . Prove that its inverse, if it exists, is also linear. Solution. We are to assume that there exists a map M such that L(x) = y if and only if M (y) = x.
Let x and x′ be two arbitrary vectors in IRn and let s and t be two arbitrary numbers. Define u = M (x) and u′ = M (x′ ). By hypothesis, u = M (x) ⇒ x = L(u) and u′ = M (x′ ) ⇒ x′ = L(u′ ). As L is linear L(su + tu′ ) = sx + tx′ , which in turn implies M (sx + tx′ ) = su + tu′ = sM (x) + tM (x′ ). Thus,
M is linear .


3 2 p
23) Suppose that det  0 p 1  = 10. What are the possible values of p?
1 0 2
Solution. Expanding along the last row


3 2 p
2 p
3 2 det  0 p 1  = det
+ 2 det p 1
0 p
1 0 2
= (2 − p2 ) + 2(3p − 0) = −p2 + 6p + 2

For this to be 10
−p2 + 6p + 2 = 10 ⇐⇒ p2 − 6p + 8 = 0 ⇐⇒ p = 2, 4 c Joel Feldman. 2011. All rights reserved.

March 7, 2011

Matrices

46

24) Let




1 3 5 ∗
0 4 0 6
A=

0 1 0 2
3 ∗ 7 8 where the ∗’s denote unknown entries. Find all possible values of det A.
Solution.




1 3 5 ∗
1 3 5 ∗
(1)
(2)
0 4 0 6
0 4 0 6 det 
 = det 

0 1 0 2
0 1 0 2
(3)
3 ∗ 7 8
0 ∗ −8 ∗ (4) − 3(1)




4 0 6
0 0 −2 (1) − 4(2)
= det  1 0 2  = det  1 0
2 
(2)
∗ −8 ∗
∗ −8 ∗
(3)
= −2 det

for all values of the ∗’s.

1
∗

0
−8

25) Suppose that the 3 × 3 matrix A obeys
Solution.

4 0 det(4A) = det  0 4
0 0

= 16

det A = 5. Compute (a) det(4A) (b) det(A2 ) (c) det(4A2 )
 

0
4 0
0  A = det  0 4
4
0 0

det(A2 ) = det A det A = 5 × 5 = 25


0
0  det A = 43 × 5 = 320
4

det(4A2 ) = det(4A) det A = 320 × 5 = 1600
26) Suppose that the 6 × 6 matrix A obeys A4 = 2A. Find all possible values of det A.
Solution.
A4 = 2A =⇒ det(A4 ) = det(2A) =⇒ (det A)4 = det(2I) det A
As in the last question det(2I) = 26 = 64. So
0 = (det A4 ) − 64 det A = det A (det A)3 − 64

Consequently det A = 0 or (det A)3 = 64 or det A = 0, 4 , assuming that A has real matrix entries.
27) Evaluate



Solution.

1
a
det  2 a a3

a a2 a3
1

a2 a3 1 a  a3 1 

a
2
a






1 a a2 a3
1
a a2 a3
(1)
2
3
4
1 
0
0
1 − a  (2) − a(1)
a a a
0
det  2
= det 

 a a3 1 a
0
0
1 − a4 a − a5 (3) − a2 (1)
3
2
4
5
2
6 a 1 a a
0 1−a a−a a −a
(4) − a3 (1)


0
0
1 − a4
0
1 − a4
= det  0
1 − a4 a − a5  = (1 − a4 ) det
4
1 − a a − a5
1 − a4 a − a5 a2 − a6
= (1 − a4 ) − (1 − a4 )(1 − a4 )

= −(1 − a4 )3 = (a4 − 1)3

We expanded along the first row to achieve each of the the second, third and fourth equalities. c Joel Feldman. 2011. All rights reserved.

March 7, 2011

Matrices

47