TUTOR CLASS
Content: Covered all chapter 3, 4, 5.
No:
* Linear Approximation * Linear programming * Leontief model

Chapter 3: INTRODUCTION TO LINEAR EQUATIONS ANDMATRICES
(Gaussian elimination) 1. Solve the following system x1-4x2+x3=5-2x1-5x2+3x3=33x1+2x2-x3=1 Using:
A, Gauss elimination method
B, Cramer’s rule. (chapter 4)

Chapter 4: EIGENVECTORS AND EIGENVALUES (Determinants, Cramer’s rule, Matrix Inverse, Eigenvectors, Eigenvalues) 2. Let A b the matrix defined by
A=3 4 -1 04 -1 0 3-1 1 2 3

A, Applying the row operation 4 R4+R2→R2 to A, what is the resulting matrix B we get?

B, Use row operations to transform A to the form C where C=0 a b c0 d e f0 g h i-1 1 2 3

And where the letters a, b, c, d, e, f, g, h, g and I are the numbers that you have to find out (depending on your row operations). Find a relation between the determinant of A and determinant of C. C, Evaluate the determinant of A, detA, by using the properties of determinants and cofactor expansions along rows and columns of your choice. 3. ( Matrix Inverse) Recall that if a matrix A is invertible, then the unique solution of the linear system of equations AX= b is given by X= A-1b Given the linear system of equations

x+2y-9z =1-2x-4y+19z=0-y+2z =1 A, Find the inverse of the matrix A of coefficient of the linear system above, where A= 1 2 -9-2 -4 190 -1 -2 4. (Eigenvectors, eigenvalues) Let
A= 2 3 04 3 00 0 6

A, Find all the eigenvalue of the matrix A. B, Given that γ1=-1 is an eigenvalue of A, find all the eigenvectors corresponding to γ1.

Chapter 5: DIFFERENT EQUATIONS (Separate differential equations, linear first-order differential equations) 5. Find the special solution organization the equation dydx= (x+2)2ey Satisfying y1=0

6. Find the solution of the initial value problem y'+2xy=xe-x2 y0=1 Solve the problem.