MODULE 5
Mathematical Tools: Determinants and Matrices
TEACHING SUGGESTIONS
Teaching Suggestion M5.1: Why Discuss Matrices and Determinants?
Teaching Suggestion M5.2: Matrices Can Be Used to Display Data in Tabular Form.
As seen in Section M5-2 in this module, a matrix is also a useful tool for presenting data.
SOLUTIONS TO PROBLEMS
M5-2. One way to calculate the determinant of a matrix is to draw the primary and secondary
M5-3. Determinants are used in solving a system of equations and in finding the inverse of a
matrix.
M5-4. A matrix of cofactors is developed as follows:
1. Select an element in the original matrix.
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
6. Return to step 1 and continue until all elements in the original matrix have been replaced
by their cofactor values.
M5-6. The inverse of a matrix may be found by dividing each term in the adjoint by the deter-
minant of the original matrix.
M5-7. a. Value = (6)(2) – (–5)(3) = 27
b. Value = (3)(–1)(–2) + (7)(2)(4)
M5-8.
( )( )( ) ( )( )( ) ( )( )( )
423
4 3 2 2 1 3 3 2 1
2 3 1
++
( )( )( ) ( )( )( ) ( )( )( )
5 4 3
5 2 2 4 1 3 3 2 3
2 2 1
++
( )( )( ) ( )( )( ) ( )( )( )
5 2 3 2
2 3 1
3 1 2
7
Z
==
=
M5-9.
5 2 3 4
2 3 1 2
3 1 2 3
X
Y
Z
 
 
=
 
 
 
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
 
=+=
 
 
d. Matrix C + matrix A: cannot be added; matrix A is (2 3) and matrix C is (3 3).
M5-11. a. B C is 2 3
b. C B is not possible.
M5-12. a. Matrix A Matrix B = Matrix C
( )
2 6 8 10
345
1 3 4 5
 
=
 
 
c. Matrix R Matrix S = Matrix T
2 3 1 0 2 0 0 3 2 3
1 4 0 1 1 0 0 4 1 4
++
   
 = =
   
++
   
d. Matrix W Matrix Y = Matrix Z
M5-13. Job matrix Cost matrix
$1
50 100 10 20 $2
70 80 20 30 $3
20 50 30 10 $5











M5-14.
Transpose of matrix
6 1 6 3
8 0 4 1
2 5 3 2
2 7 1 7
R



=


M5-15.
Element
Determinant
Value of
Removed
of Cofactors
Cofactor
Row 1, column 1
08 48
69
=−
–48
Row 1, column 2
Row 1, column 3
12
Row 2, column 1
28 6
=−
+6
69
=−
Row 2, column 2
17 12
39
=−
–12
Row 2, column 3
Row 3, column 1
Row 3, column 2
28
=−
Row 3, column 3
14 8
20
=−
–8
Matrix of cofactors
48 6 12
6 12 6
32 6 8
−


=−


−

M5-16.
Original matrix
1 4 7
208
369


=


48/ 60 6/ 60 32/ 60
6/ 60 12/ 60 6/60
12/60 6/ 60 8/ 60
−


=−


−

To verify, we multiply the original matrix by its inverse. An identity matrix indeed results.