Question

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

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

Answer 1

Let A = `[[2,-2],[4,3]]`

|A|= `|[2,-2],[4,3]|` = 6 + 8 = 14 `≠` 0

∴  A⁻¹ exist

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 = 3

A₁₂ = (− 1)¹⁺²M¹² = − 4

A₂₁ = (− 1)²⁺¹M₂₁ = (− 2) = 2

A₂₂ = (− 1)²⁺²M₂₂ = 2

Hence, the co-factor matrix

 = `[("A"_11,"A"_12),("A"_21,"A"_22)]` = `[(3,-4),(2,2)]`

∴ adj A `=[(3,2),(-4,2)]`

∴ A⁻¹ = `1/|"A"|` (adj A) = `1/14[(3,2),(-4,2)]`