Question

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

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

Answer 1

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

∴ |A| = `|(-1,5),(-3,2)|` = − 2 + 15 = 13 `≠` 0

∴ A⁻¹ exists.

First we have to find the co-factor matrix

= [A_ij]_2×2′ where A_ij = (− 1)^i+jM_ij

Now, A₁₁ = (− 1)¹⁺¹M₁₁ = 2

A₁₂ = (− 1)¹⁺²M₁₂ = − (− 3) = 3

A₂₁ = (− 1)²⁺¹M₂₁ = − 5

A₂₂ = (− 1)²⁺²M₂₂ = − 1

Hence, the co-factor matrix

= `[("A"_11,"A"_12),("A"_21,"A"_22)]` = `[(2,3),(-5,-1)]`

∴ adj A = `[(2,-5),(3,-1)]`

∴ A⁻¹ = `1/|"A"|` (adj A) = `1/13[(2,-5),(3,-1)]`