Matrix Calculations Using Calculators
The calculators most commonly used by Foundation Studies students at UTAR are Casio fx-350MS, fx-570MS, and fx-570ES. Casio fx-350MS has no matrix function and cannot be used to perform matrix calculation. Both Casio fx-570MS and Casio fx-570ES have matrix mode. Casio fx-570ES can display fully a 3×3 matrix with all its 9 elements visible. Casio fx-570MS has only a 2-line display and can only show matrix elements one at a time. Hence the operations for matrix computation are different for these two series of calculators. The general procedures in matrix calculations are as follows: (1) Enter Matrix mode of the calculators. (2) Assign a variable to store the matrix. There are 3 variables available: MatA, MatB, and MatC. (3) Select the dimension or order (1×1 to 3×3) of the matrix. (4) Input the elements of the matrix; the data will be automatically saved in the matrix variable assigned. (5) Exit the matrix input or edit mode by pressing the coloured AC key. (6) Press SHIFT MATRIX or SHIFT MAT (Matrix function is at numeric key 4) to recall the stored matrix and perform matrix calculations as needed.

(a) Casio fx-570ES
(a) Entering a matrix. 1. 2. 3. 4. Press the MODE key. Select 6:Matrix mode. A display Matrix? will be shown to let us select one of the 3 possible matrix variables allowable. Let us choose 1:MatA by pressing 1. The next display permits us to select the order (m×n) of the matrix MatA. Let us choose 1: 3×3 by pressing 1. An input screen will be displayed for us to key in the elements of the matrix. Let us key in the following matrix as an exercise:
1 1 2    4 0 3  5 1 2  

Press 1=, 1=, 2=, 4=, 0=, 3=, 5=, 1=, and 2=. (The = key acts like the Enter key on a computer keyboard.) 5. Exit from the matrix input screen by pressing the AC key. (The AC acts like the ESC key on a computer keyboard.)

(b) Finding the inverse of a non-singular 3×3 matrix. 6. Next let us compute the inverse of MatA in memory that we have just keyed in. Press SHIFT MATRIX (Matrix function is at numeric key 4). Select option 3:MatA.

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7. 8.

The word MatA will be displayed. To find out its inverse, press the x-1 key. The display becomes MatA-1. Press the = key. After a short delay, the inverse of the matrix will be displayed:
 1 − 4   7  12   1   3 0 − 2 3 1 3 1  4   5  12   1 −   3

(There will ne no inverse if the matrix is singular, i.e. its determinant is zero.) 9. The result of this matrix calculation, i.e. the inverse of the matrix, will be stored in the memory variable MatAns. We can do matrix calculation with MatAns. For example, we can multiply MatA with MatAns to get the identity matrix. Press SHIFT MATRIX. Select option 6:MatAns. MatAns will appear on the screen. Press the multiplication key ×. Press SHIFT MATRIX. Select 3: MatA. MatAns×MatA will appear on the screen. Press = to display the identity matrix:
1 0 0   0 1 0 0 0 1  

10. 11.

(c) Finding the determinant of a 3×3 matrix. 12. 13. 14. We can also compute the determinant of matrix MatA. Press SHIFT MATRIX. Select 7:det. The screen will display det(. Select the matrix whose determinant is to be determined. Press SHIFT MATRIX. Choose 3:MatA by pressing 3. The display becomes det(MatA. Close the bracket by pressing ). Press = to display the determinant. The determinant of value 12 will be displayed.

(d) Finding the transpose of a matrix. 15. 16. 17. We can also display the transpose of the matrix easily. Press SHIFT MATRIX. Choose 8: Trn. The screen will display Trn(. Tell the calculator the matrix we want to transpose. Press SHIFT MATRIX. Choose 3: MatA by pressing key 3. Close the bracket by pressing ), followed by =. The transpose of MatA will be displayed.

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(e) Modifying the content of a matrix. 18. 19. To modify the matrix MatA already in memory, press SHIFT MATRIX. Choose 2:Data. The screen will display Matrix? 1: MatA 2: MatB 3: MatC. Since our matrix is stored in MatA, choose 1: MatA by pressing 1. The matrix A will be displayed. We can alter the values of the elements now. The changed values will be stored automatically when we exit from the edit display. Exit from the matrix edit screen by pressing the AC key.

20. 21.

(f) Multiplying one matrix by another matrix. 22. 23. 24. To multiply MatA already in memory by another memory, assign a variable to the second matrix first. Press SHIFT MATRIX. Select 2: Data. The screen will display Matrix? 2: MatB 3: MatC. 1: MatA Let us assign the second matrix to variable MatB. Select 2: MatB by pressing 2. The input screen for MatB will be displayed. As an exercise, enter the following matrix data:
2 3 1    5 7 3  4 0 6  

25.

26. 27. 28.

Exit from the matrix edit screen by pressing the AC key. Now we can call MatA and MatB and multiply them together. Press SHIFT MATRIX. Select 3: MatA. MatA will appear on the screen. Press the multiplication key ×. Press SHIFT MATRIX. Select 4: MatB. MatA×MatB will appear on the screen. Press = to display the result:
 15 10 16     20 12 22   23 22 20   

(g) Solving system of linear equations in three variables. 29. Suppose we are required to solve the following system of linear equations: 2 x + 3 y + 3z = 0 4x − y − 6z = 1 − 4x + 3y + 6z = 3 3/13

The equations can be expressed in matrix form as:  2 3 3  x   0        4 −1 −6   y  =−1   4 3 6  z   3       The values of x, y, z are given by −1  x 2 3 3   0         y =  4 −1 −6   −1  z  4 3 6   3        30. We now enter the values of the matrices. Press SHIFT MATRIX. Select 2: Data. Assign the first matrix to variable MatA and select the order of the matrix MatA as 3×3. Note: If MatA is already in memory, selecting 2: Data would bring up the already defined MatA and we can edit the matrix elements. If MatA is not yet in memory, then we need to assign MatA to the new matrix by selecting the order of the matrix. If MatA is already in memory and its order is not what we wanted, then we need to start from scratch - press the MODE key, select 6:Matrix mode, choose 1:MatA, and define the order. 31. An input screen will be displayed for us to key in the elements of the MatA:
2 3 3     4 −1 −6  4 3 6   

32. 33. 34.

Exit from the matrix input screen by pressing the AC key. Next we create a second matrix. Press SHIFT MATRIX. Select 2: Data. Assign the second matrix to variable MatB and select the order of MatB as 3×1. The input screen for MatB will be displayed. Enter the following matrix data:
0    −1 3  

35. 36.

Exit from the matrix edit screen by pressing the AC key. Now we are ready to find the solutions to the system of linear equations. Compute the inverse of MatA in memory that we have keyed in. Press SHIFT MATRIX. Select option 3:MatA. The word MatA will be displayed. To determine its inverse, press the x-1 key. The display becomes MatA-1. Press the = key. After a short delay, the inverse of MatA will be displayed: 4/13

37. 38.

1 5   1 − 6 8 24    1  2 0 −  3 3   7  − 2 − 1   12 36   9 The inverse matrix will be stored in memory as MatAns.

39. 40. 41.

Exit from the matrix edit screen by pressing the AC key. Recall from memory the inverse matrix we have just computed. Press SHIFT MATRIX. Select 6: MatAns. MatAns will appear on the screen. Press the multiplication key ×. Press SHIFT MATRIX. Select 4: MatB. MatAns×MatB will appear on the screen. Press = to display the result: 1 2    −1 2   3  x    y = z   1 2    −1 2   3

42.

Hence,

Note: (i)

The defined matrix in memory can be cleared by changing the entry mode from matrix back to general computation: Press the MODE key. Select 1:COMP. Please note that using SHIFT CLR (key 9) does not clear the memory of the matrix entered.

(ii)

If seeing the data on the display is a problem, change the contrast in display using the following steps: Press Shift Setup, followed by the navigation key  to display the second screen. Choose 6: Cont. Press the keys [Light] and [Dark] to change the display contrast as desired. Press AC to exit contrast adjustment.

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(b) Casio fx-570MS
(a) Entering a matrix. 1. Press the MODE key three times. The displays are as follows: First time : COMP CMPLX Second time : SD REG BASE Third time : EQN MAT VCT 1 2 3 2. Press 2 to select Matrix mode. A tiny MAT indicator will be visible at the central topmost row of the display. 3. Press SHIFT MAT (Matrix function is at numeric key 4). The screen will display : Dim Edit Mat 1 2 3 Press 1 to select the order (or dimension) of the matrix. Always define the dimension of the matrix first when creating a new matrix. 4. The next display will show A B C 1 2 3 A, B, C stands for three possible matrix variables: MatA, MatB, MatC. Choose MatA by pressing 1. 5. The next display will be MatA (m×n) m? Press 3 followed by = to define the row number of a 3×3 matrix. (The = key functions like the Enter key on a computer keyboard.) 6. The next display will be MatA (m×n) n? Press 3 followed by = to define the column number of a 3×3 matrix As an exercise, key in the matrix below.
1 1 2    4 0 3  5 1 2  

7. The next display is MatA11. Key in 1 followed by = to define the element for row 1 and column 1. 8. The next display is MatA12. Key in 1 followed by = to define the element for row 1 and column 2. 9. The next display is MatA13. Key in 2 followed by = to define the element for row 1 and column 3. 10. The next display is MatA21. Key in 4 followed by = to define the element for row 2 and column 1.

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11. The next display is MatA22. Key in 0 followed by = to define the element for row 2 and column 2. 12. The next display is MatA23. Key in 3 followed by = to define the element for row 2 and column 3. 13. The next display is MatA31. Key in 5 followed by = to define the element for row 3 and column 1. 14. The next display is MatA32. Key in 1 followed by = to define the element for row 3 and column 2. 15. The next display is MatA33. Key in 2 followed by = to define the element for row 3 and column 3. 16. This will bring us back to the beginning of the matrix MatA11. We may use the navigation key to scroll through all the elements for checking purpose. Press AC to exit matrix input mode. (The AC key functions like the ESC key on a computer keyboard – it clears the display screen.) (b) Finding the inverse of a non-singular 3×3 matrix. 17. Press SHIFT MAT (Matrix function is at numeric key 4). The screen will display : Dim Edit Mat 1 2 3 Press 3 to select the 3×3 matrix just entered that is stored in memory. 18. The next display will be A B C 1 2 3 A, B, C stands for three matrices: MatA, MatB, MatC. Choose MatA by pressing 1. 19. The word MatA will be displayed. To find out its inverse, press the x-1 key. The display becomes MatA-1. Press = to find the inverse of the matrix, which is stored in variable MatAns. 20. The display becomes MatAns11, the row 1, column 1 of the inverse matrix element, the value of which is displayed at the second row of the screen. By default, the values b/c are in decimal form. Press the a key to change it into a fraction, if necessary. 21. Use the navigation key  to move to the next column. The display is MatAns12 with the value at the second row of the screen. 22. Repeat pressing the navigation keys   to display all the elements of the inverse matrix, an element at a time.
 1 − 4   7  12   1   3 0 − 2 3 1 3 1  4   5  12   1 −  3

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(There will ne no inverse if the matrix is singular, i.e. its determinant is zero.) 23. Press AC to exit from the matrix display mode. 24. The result of this matrix calculation, i.e. the inverse of the matrix, will be stored in the memory variable MatAns. We can do matrix calculation with MatAns. For example, we can multiply MatA with MatAns to get the identity matrix. 25. Press SHIFT MAT. Press 3 to select a matrix in memory. 26. The next display will be A B C Ans 1 2 3 4 The symbols stands for the four possible matrices: MatA, MatB, MatC, and MatAns. Choose MatAns by pressing 4. MatAns will appear on the screen. 27. Press the multiplication key ×. Press SHIFT MAT. Press 3 to select a matrix. 28. Press 1 to select MatA. MatAns×MatA will appear on the screen. Press = to display the identity matrix, again one element at a time.
1 0 0   0 1 0 0 0 1  

(c) Finding the determinant of a 3×3 matrix. 29. Press SHIFT MAT (Matrix function is at numeric key 4). The screen will display : Dim Edit Mat 1 2 3 30. Use the navigation key  to scroll to the right: Det Trn 1 2 Press 1 to select Det. 31. The display will next show Det _ . Load the stored MatA by pressing SHIFT MAT. The screen will display : Dim Edit Mat 1 2 3 Press 3 to call the 3×3 matrix just entered. 32. The next display will show A B C Ans 1 2 3 4 A, B, C stands for three possible matrices: MatA, MatB, and MatC. Choose MatA by pressing 1. 33. The display will become Det MatA. Press =. The value of the determinant will be displayed at the second screen row.

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(d) Finding the transpose of a matrix. 34. Press SHIFT MAT. The screen will display : Dim Edit Mat 1 2 3 35. Use the navigation key  to scroll to the next screen: Det Trn 1 2 Press 2 to select Trn. 36. The screen will display Trn _. Press SHIFT MAT. 37. The screen will display : Dim Edit 1 2 Press 3 to select a matrix. Mat 3

38. The next display will show A B C Ans 1 2 3 4 A, B, C stands for three possible matrices: MatA, MatB, and MatC. Choose MatA by pressing 1. The screen will display Trn MatA. Press = key. 39. The first element of the transpose of MatA MatAns11 will be displayed. Use the navigation keys  or the = key to show the required element of the transposed matrix. (e) Modifying the content of a matrix. 40. Press SHIFT MAT. The screen will display : Dim Edit Mat 1 2 3 Select Edit by pressing 2. 41. The next display will show A B C Ans 1 2 3 4 A, B, C stands for three possible matrices: MatA, MatB, and MatC. Choose MatA by pressing 1. 42. The first element of MatA MatA11will be displayed. Change the value of the element as required. We may use the navigation keys  to move to the required element. (f) Multiplying one matrix by another matrix. 43. To multiply MatA already in memory by another memory, assign a variable to the second matrix by first defining its dimension.

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44. Press SHIFT MAT (Matrix function is at numeric key 4). The screen will display : Dim Edit Mat 1 2 3 Select Dim by pressing 1 to define the order of the second matrix. 45. The next display will show A B C Ans 1 2 3 4 Select B to assign the second matrix to MatB. Press 2. 46. The next display will be MatB (m×n) m? Press 3 followed by = to define the row number of a 3×3 matrix. The key = functions like the Enter key on a computer keyboard. 47. The next display will be MatB (m×n) n? Press 3 followed by = to define the column number of a 3×3 matrix. As an exercise, key in the following second matrix:
2 3 1    5 7 3  4 0 6  

48. The next display is MatB11. Key in 2 followed by = to define the element for row 1 and column 1 of MatB. 49. The next display is MatB12. Key in 3 followed by = to define the element for row 1 and column 2. 50. Similarly key in all the remaining elements of MatB : MatB13, MatB21, MatB22, MatB23, MatB31, MatB32, and MatB33. 51. This will bring us back to the beginning of the matrix MatB11. Exit from the matrix input mode by pressing AC. 52. Now we can call MatA and MatB and multiply them together. Press SHIFT MAT. The screen will display : Dim Edit Mat 1 2 3 Select Mat by pressing 3. 53. The next display will show A B C 1 2 3 Select A to load in the matrix MatA. Press 1. Ans 4

54. MatA will appear on the screen. Press the multiplication key ×.

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55. Press SHIFT MAT. The screen will display : Dim Edit Mat 1 2 3 Select Mat by pressing 3. 56. The next display will show A B C 1 2 3 Select B to load in the matrix MatB. Press 2. 57. MatA×MatB will appear on the screen. Press =. 58. The first element of the matrix product MatAns11 will be displayed. Use the navigation keys  to display the rest of the elements: MatAns12, MatAns13, MatAns21 MatAns22, MatAns23, MatAns31, MatAns32, and MatAns33. The resulting matrix should be:
 15 10 16     20 12 22   23 22 20   

Ans 4

(g) Solving system of linear equations in three variables. 59. Suppose we are required to solve the following system of linear equations: 2 x + 3 y + 3z = 0 4x − y − 6z = 1 − 4x + 3y + 6z = 3 The equations can be expressed in matrix form as:  2 3 3  x   0        4 −1 −6   y  =−1   4 3 6  z   3       The values of x, y, z are given by

 x    y = z  

2 3 3   0       4 −1 −6   −1 4 3 6   3     

−1

60. We now enter the values of the matrices. Press SHIFT MATRIX. The screen will display : Dim Edit Mat 1 2 3 Select Edit by pressing 2. 61. The next display will show A B 1 2 Choose MatA by pressing 1. C 3 Ans 4

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62. Since we have previously defined MatA, the first element of MatA MatA11 will be displayed. Replace MatA11 with 2. Press the navigation keys  to move to MatA12 and replace it with 3. Continue using navigation keys to modify the matrix MatA as
2 3 3     4 −1 −6  4 3 6   

63. Exit from the matrix input screen by pressing the AC key. 64. Next we create a second matrix. Press SHIFT MAT. The screen will display : Dim Edit Mat 1 2 3 Select Dim by pressing 1 to define the order of the new matrix. 65. The next display will show A B C Ans 1 2 3 4 Select B to assign the second matrix to MatB. Press 2. 66. The next display will be MatB (m×n) m? Press 3 followed by = to define the row number of a 3×1 matrix. 67. The next display will be MatB (m×n) n? Press 1 followed by = to define the column number of a 3×1 matrix. 68. The next display is MatB11. Enter 0 for MatB11, −1 for MatB21, 3 for MatB31 as in
0    −1 3  

69. Exit from the matrix edit screen by pressing the AC key. 70. Now we are ready to find the solutions to the system of linear equations. Compute the inverse of MatA that we have keyed in. Press SHIFT MATRIX. Select option 3:Mat., and then 1: A. 71. The word MatA will be displayed. To determine its inverse, press the x-1 key. The display becomes MatA-1. 72. Press the multiplication key ×. Press SHIFT MATRIX. Select 3: Mat. , then select matrix 2: B. The first element of MatA-1×MatB, MatAns11, will appear on the screen. Use the navigation keys  to display the other elements. The resulting matrix b/c MatAns should be (the results may be in decimal values; press the a key to change it into a fraction, if necessary):

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1 2    −1 2   3  x    y = z   1 2    −1 2   3

73. Hence,

Note: The defined matrix in memory will be cleared by pressing the Shift CLR key.

Leong Sow Chew Lecturer Centre of Foundation Studies, Universiti Tunku Abdul Rahman (Kampar Campus)

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