466
Relations, Functions, and Matrices
40. Use the method of Exercise 39 to find the inverse of matrix a in Example 71.
41. Consider a system of n linear equations in n unknowns, such as the one from Example 63:
x + y = 70
24x + 14y = 1180
If
a = c
1
1
24 14
d
X = c
x
y
d
b = c
70
1180
d
then the system of equations can be represented in matrix form by
a # X = b
If a, the matrix of coefficients, is invertible, then we can multiply both sides of the above equation by
a
−1
, giving
a
−1 # (a # X) = a
−1 # b
(a
−1 # a) # X = a
−1 # b
(matrix multiplication is associative)
i # X = a
−1 # b
(definition of a
−1
)
X = a
−1 # b
(definition of i)
Therefore the solution to the system of equations is given by
X = a
−1 # b
Make use of Exercise 39 to find a
−1
, and use this approach to solve the system of equations.
In Exercises 42–46, solve the systems of equations using the method of Exercise 41.
42. x + 2y = −4
x + y = 5
43. The system of Exercise 27
44. The system of Exercise 28
45. The system of Exercise 29
46. The system of Exercise 30
47. For Boolean matrices
a = £
1 0 0
1 1 0
0 1 1
§
b = £
1 0 1
0 1 1
1 1 1
§
find a ` b, a ~ b, a × b, and b × a.