Question

Find the inverse of the following matrix.

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

Answer 1

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

∴ |A| = `|(1,2),(2,-1)|` = − 1 − 4 = − 5 `≠` 0

∴ A⁻¹ exists.

consider AA⁻¹ = I

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

By R₂ − 2R₁, we get

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

By `(-1/5)`R₂, we get, 

`[(1,2),(0,1)]`A⁻¹ = `[(1,0),(2//5,-1//5)]`

By R₁ − 2R₂, we get,

`[(1,0),(0,1)]`A⁻¹ = `[(1//5,2//5),(2//5,-1//5)]`

∴ A⁻¹ = `-1/5[(-1,-2),(-2,1)]`

∴ A⁻¹ = `1/5[(1,2),(2,-1)]`

The answer can be checked by finding the product AA⁻¹.

AA⁻¹ = `[(1,2),(2,-1)][(1//5,2//5),(2//5,-1//5)]`

= `[(1(1/5) + 2(2/5),1(2/5) + 2(-1/5)),(2(1/5) - 1(2/5),2(2/5) -1(-1/5))]`

= `[(1/5 + 4/5,2/5 - 2/5),(2/5 - 2/5,4/5 + 1/5)] = [(1,0),(0,1)]` = I

Hence, A⁻¹ is the required answer.