Question

If `A = [(0, 2, 1), (-2, -1, -2), (1, -1, 0)]`, find A⁻¹ and use it to solve the following system of equations:

−2y + z = 7, 2x − y − z = 8, x − 2y = 10

Answer 1

`A = [(0, 2, 1), (-2, -1, -2), (1, -1, 0)]`

|A| = 0(−1 − 2) − 2(0 − (−2)) + 1(2 − (−1))

|A| = 0 − 4 + 3 = −1

⇒ Cofactor matrix:

`C = [(-2, -2, 3), (-2, -1, 2), (-3, -2, 4)]`

adj A = `C^T = [(-2, -2, 3), (-2, -1, 2), (-3, -2, 4)]`

`A^-1 = 1/|A|adjA`

`A^-1 = 1/-1 [(-2, -2, 3), (-2, -1, 2), (-3, -2, 4)]`

`A^-1 = [(2, 1, 3), (2, 1, 2), (-3, -2, -4)]`

⇒ Given equations:

−2y + z = 7

2x − y − z = 8

x − 2y = 10

∴ The coefficient matrix is `[(0, -2, 1), (2, -1, -1), (1, -2, 0)] = A^T`

So, A^TX = B

X = (A^T)⁻¹B

X = (A⁻¹)^TB

`(A^-1)^T = [(2, 2, -3), (1, 1, -2), (3, 2, -4)]`

`X = [(2, 2, -3), (1, 1, -2), (3, 2, -4)] [(7), (8), (10)]`

`X = [(14 + 16 - 30), (7 + 8 - 20), (21 + 16 - 40)]`

∴ `X = [(0), (-5), (-5)]`

Hence,

x = 0, y = −5, z = −3