Question

Find the inverse of the following matrix by the adjoint method.

`[(1, 0, 0),(3, 3, 0),(5, 2, -1)]`

Answer 1

Let A = `[(1, 0, 0),(3, 3, 0),(5, 2, -1)]`

∴ |A| = `|(1, 0, 0),(3, 3, 0),(5, 2, -1)|`

= 1(−3 − 0) − 0 + 0

= −3 ≠ 0

∴ A⁻¹ exist

First we have to find the co-factor matrix

= [A_ij]_3×3′ where A_ij = (− 1)^i+jM_ij

Now, A₁₁ = (−1)¹⁺¹M₁₁ = 1`|(3, 0),(2, -1)|` = 1(−3 − 0) = −3

A₁₂ = (−1)¹⁺²M₁₂ = − 1`|(3, 0),(5, -1)|` = −1(−3 − 0) = 3

A₁₃ = (−1)¹⁺³M₁₃ = 1`|(3, 3),(5, 2)|` = 1(6 − 15) = −9

A₂₁ = (−1)²⁺¹M₂₁ = −1`|(0, 0),(2, -1)|` = −1(0 − 0) = 0

A₂₂ = (−1)²⁺²M₂₂ = 1`|(1, 0),(5, -1)|` = 1(−1 − 0) = −1

A₂₃ = (−1)²⁺³M₂₃ = −1`|(1, 0),(5, 2)|` = −1(2 − 0) = −2

A₃₁ =(−1)³⁺¹M₃₁ = 1`|(0, 0),(3, 0)|` = 1(0 − 0) = 0

A₃₂ = (−1)³⁺²M₃₂ = −1`|(1, 0),(3, 0)|` = −1(0 − 0) = 0

A₃₃ = (−1)³⁺³M₃₃ = 1`|(1, 0),(3, 3)|` = 1(3 − 0) = 3

∴ The co-factor matrix

= `[(A_11, A_12, A_13),(A_21, A_22, A_23),(A_31, A_32, A_33)]` = `[(-3, 3, -9),(0, -1, -2),(0, 0, 3)]`

∴ adj A = `[(-3, 0, 0),(3, -1, 0),(-9, -2, 3)]`

∴ A⁻¹ = `1/|A|` (adj A)

= `-1/3[(-3, 0, 0),(3, -1, 0),(-9, -2, 3)]`

∴ A⁻¹ = `1/3[(3, 0, 0),(-3, 1, 0),(9, 2, -3)]`