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M O D U L E
Mathematical Tools: Determinants and Matrices
5
TEACHING SUGGESTIONS
Teaching Suggestion M5.1: Why Discuss Matrices and Determinants? These mathematical tools are needed in Markov analysis to compute absorbing states. In a more advanced treatment of linear programming, matrix manipulation skills are also desired. Teaching Suggestion M5.2: Matrices Can Be Used to Display Data in Tabular Form. As seen in Section M5-3 in this module, a matrix is also a useful tool for presenting data.
M5-7.
a. Value b. Value
(6)(2)
( 5)(3)
27 (4)( 1)( 6)
(3)( 1)( 2) ( 6)(1) (3) (3)(2)(3) 6 56 18
(7)(2)(4) ( 2)(1)(7) 24 18 14 16
M5-8. (4)(3)(2) - (3)(3)(3) X= = (5)(3)(2) 5 2 3 2 3 1 - (3)(3)(3) 3 1 2 -3 = 2 5 4 3 2 2 1 3 3 2 4 2 3 2 3 1 3 1 2 + - + - (2)(1)(3) (1)(1)(4) (2)(1)(3) (1)(1)(5) + - + - (3)(2)(1) (2)(2)(2) (3)(2)(1) (2)(2)(2)
SOLUTIONS TO PROBLEMS
M5-1. Matrices are used in Markov analysis and game theory. They may also be used to represent a system of equations in linear programming. M5-2. One way to calculate the determinant of a matrix is to draw the primary and secondary diagonals. The numbers in each diagonal are multiplied. Each of the products found on a primary diagonal are added, and each of the products found on a secondary diagonal are subtracted. The resulting number is the determinant. M5-3. Determinants are used in solving a system of equations and in finding the inverse of a matrix. M5-4. 1. A matrix of cofactors is developed as follows: Select an element in the original matrix.
(5)(2)(2) + (4)(1)(3) + (3)(2)(3) - (3)(2)(3) - (3)(1)(5) - (2)(2)(4) Y= = 2 5 2 3 2 3 1 3 1 2 = 5 2 4 2 3 2 3 1 3 1 2
2. Draw a line through the row and column of the element selected. The numbers uncovered represent the cofactor for that element. 3. Calculate the value of the determinant of the cofactor. 4. Add together the location numbers of the row and column crossed out in step 2. If the sum is even, the sign of the determinant's value from step 3 does not change. If the sum is an odd number, change the sign of the determinant's value. 5. The number just computed becomes an entry in the matrix of cofactors; it is located in the same position as the element selected in step 1. 6. Return to step 1 and continue until all elements in the original matrix have been replaced by their cofactor values. M5-5. The adjoint of a matrix is the transpose of the matrix of cofactors. M5-6. The inverse of a matrix may be found by dividing each term in the adjoint by the determinant of the original matrix.
(5)(3)(3) + (2)(2)(3) + (4)(2)(1) - (3)(3)(4) - (1)(2)(5) (3)(2)(2) - Z= = 2 5 2 3 2 3 1 3 1 2 7 = 2 To verify, we check to be sure that 5X 5(
3 ) 2
2Y
3Z
4; namely,
2( )
1 2
3( )
7 2
8 2
4.
M5-9. 5 2 3 X 4 2 3 1 Y = 2 3 1 2 Z 3
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M5-10. a. Matrix A matrix B 2 4 1 7 6 5 9 10 6 = = + 3 8 7 0 1 2 3 9 9 b. Matrix B 7 6 5 2 4 1 matrix A = - 0 1 2 3 8 7 5 2 4 = -3 -7 -5 c. Matrix C matrix D 3 6 9 5 1 6 8 7 15 = 7 8 1 + 4 0 6 = 11 8 7 9 2 4 3 1 5 12 3 9 d. Matrix C matrix A: cannot be added; matrix A is (2 3) and matrix C is (3 3). Only matrices of the same dimension can be added or subtracted. M5-11. a. b. c. d. e. B C A C D C is 2 3 B is not possible. B is not possible. D is 3 C is 3 3. 3. Matrix B Matrix C 6 8 10 3 4 5 Matrix G M5-15.
$50 + 200 + 30 + 100 $380 = $70 + 160 + 60 + 150 = $440 $20 + 100 + 90 + 50 $260 Cost of dormitory job is $380, of office building job is $440, and of apartment complex job is $260. M5-14. 6 8 Transpose of matrix R = 2 2 1 0 5 7 6 4 3 1 3 1 2 7
3 2 5 Transpose of matrix S = 1 2 4
Element Removed
Determinant of Cofactors
Value of Cofactor
Row 1, column 1
0 8 = -48 6 9 2 8 = -6 3 9 2 0 = 12 3 6 4 7 = -6 6 9 1 7 = -12 3 9 1 4 = -6 3 6 4 7 = 32 0 8 1 7 = -6 2 8 1 4 = -8 2 0
48
M5-12. a. Matrix A 2 1 b. Matrix E
Row 1, column 2
6
(3
4 5) =
Row 1, column 3
12
Matrix F
4 3 (5 2 6 1) 2 = (20 + 6 + 12 + 0) = (38) 0 c. Matrix R Matrix S Matrix F 2 3 1 0 2 + 0 0 + 3 2 3 = = 1 4 0 1 1 + 0 0 + 4 1 4 d. Matrix W 3 5 2 1 1 4 5 1 2 3 6 5 4 4 3 + 10 12 + 15 15 + 30 3 + 25 8 + 3 10 + 6 2 + 5 = 2 + 2 4 + 8 16 + 12 20 + 24 4 + 20 13 27 45 28 7 = 4 11 16 12 28 44 24 M5-13. Job matrix 50 100 10 20 70 80 20 30 20 50 30 10 Cost matrix $1 $2 $3 $5
Row 2, column 1
6
Row 2, column 2
12
Row 2, column 3
6
Row 3, column 1
32
Row 3, column 2
6
Row 3, column 3
8
-48 6 12 Matrix of cofactors = 6 -12 6 6 -8 32 -48 6 32 Adjoint of matrix = 6 -12 6 6 -8 12
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M5-16.
1 4 7 Original matrix = 2 0 8 3 6 9 48 72
To verify, we multiply the original matrix by its inverse. An identity matrix indeed results. M5-17. 5X1 4X1 M5-18. 0X1 1X1 X2 2X2 4X2 2X2 240 320 3X3 2X3 28 16
Value of 0 96 84 0 determinant 60 -48 6 32 Adjoint calculation 6 -12 6 = in Problem M57 6 -8 12
-48 / 60 6 / 60 32 / 60 = 6 / 60 -12 / 60 6 / 60 6 / 60 -8 / 60 12 / 60