Question

Find the inverse of the matrices (if it exists).

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

Answer 1

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

Then, |A| = `|(2,-2),(4,3)|`

= 6 + 8

= 14 ≠ 0

So, A is a non-singular matrix and therefore it is invertible. Let C_ij be the cofactor of a_ij in A.

Then, the cofactors of elements of A are given by,

C₁₁ = (−1)¹⁺¹ (3) = 3

C₁₂ = (−1)¹⁺² (4) = (−4)

C₂₁ = (−1)²⁺¹ (−2) = 2

C₂₂ = (−1)²⁺² (2) = 2

∴ Adj A = `[(3,-4),(2,2)]^' = [(3,2),(-4,2)]`

Hence, A⁻¹ = `1/|A| ("Adj A")`

= `1/14 [(3,2),(-4,2)]`