Question

Find inverse of the following matrices (if they exist) by elementary transformations:

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

Answer 1

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

∴ |A| = `|(1, -1),(2, 3)|`

= 3 + 2
= 5 ≠ 0
∴ A^–1 exists.
Consider AA^–1  = I

∴ `[(1, -1),(2, 3)] "A"^-1 = [(1, 0),(0, 1)]`

Applying R₂ → R₂ – 2R₁, we get

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

Applying R₂ → `(1/5)` R₂, we get

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

Applying R₁ → R₁ + R₂, we get

`[(1, 0),(0, 1)] "A"^-1 = [(3/5, 1/5),(-2/5, 1/5)]`

∴ A^–1 = `[(3/5, 1/5),(-2/5, 1/5)]`.