Question

If A = `[(1, 2, 0), (-2, -1, -2), (0, -1, 1)]`, find A⁻¹. Using A⁻¹, solve the system of linear equations   x − 2y = 10, 2x − y − z = 8, −2y + z = 7.

If A = `[(1, 2, 0), (-2, -1, -2), (0, -1, 1)]`, then find A⁻¹.

Hence, solve the system of linear equations:

x − 2y = 10

2x − y − z = 8

−2y + z = 7

Answer 1

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

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

= 1(−1 − 2) − 2(−2 + 0) + 0

= −3 + 4 

= 1

|A| ≠ 0 A⁻¹ exist.

Now find minors and cofactors

A₁₁ = M₁₁ = −3

A₁₂ = −M₁₂ = −(−2)

= 2

A₁₃ = M₁₃ = 2

A₂₁ = −M₂₁ = −2

A₂₂ = M₂₂ = 1

A₂₃ = −M₂₃ = −(−1)

= 1

A₃₁ = M₃₁ = −4

A₃₂ = −M₃₂ = (−2)

= −2

A₃₃ = M₃₃ = (−1 + 4)

= 3

adj A = [Cofactor matrix] = `[(-3, 2, 2), (-2, 1, 1), (-4, 2, 3)]`

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

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

Given: x − 2y = 10

2x − y − z = 8

−2y + z = 7

In matrix form, `[(1, -2, 0), (2, -1, -1), (0, -2, 1)][(x), (y), (z)] = [(10), (8), (7)]`

A'X = B

X = (A')⁻¹

B = (A⁻¹)'B

`[(x), (y), (z)] = 1/1[(-3, -2, -4), (2, 1, 2), (2, 1, 3)][(10), (8), (7)]`

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

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

= `[(-30 + 30), (-20 + 15), (-40 + 37)]`

`[(x), (y), (z)] = [(0), (-5), (-3)]`

Hence, x = 0, y = −5, z = −3.