Question

Find inverse, by elementary row operations (if possible), of the following matrices

`[(1, 3),(-5, 7)]`

Answer 1

Let A = `[(1, 3),(-5, 7)]`

|A| = 1 × 7 – (– 5) × 3

= 7 + 15

= 22 ≠ 0

So, A is invertible.

Let A = IA

⇒ `[(1, 3),(-5, 7)] = [(1, 0),(0, 1)]"A"`

R₂ → R₂ + 5R₁ 

⇒ `[(1, 3),(0, 22)] = [(1, 0),(5, 1)]"A"`

`"R"_2 -> 1/22 "R"_2`

⇒ `[(1, 3),(0, 1)] = [(1, 0),(5/22, 1/22)]"A"`

R₁ → R₁ + 3R₂  

⇒ `[(1, 0),(0, 1)] = [(7/22, (-3)/22),(5/22, 1/22)]"A"`

So `"A"^-1 = [(7/22, (-3)/22),(5/22, 1/22)]`

⇒ `1/22[(7, -3),(5, 1)]`

Hence, inverse of `[(1, 3),(-5, 7)]` is `1/22 [(7, -3),(5, 1)]`