Question

If A = `[(3, 1),(5, 2)]`, and AB = BA = I, then find the matrix B

Answer 1

AB = BA = I

⇒ B = A⁻¹

|A| = `|(3, 1),(5, 2)|`

= 6 – 5

= 1

∴ A₁₁ = (–1)¹⁺¹ M₁₁ = M₁₁ = 2

A₁₂ = (–1)¹⁺² M₁₂ = – M₁₂ = – 5

A₂₁ = (–1)²⁺¹ M₂₁ = – M₂₁ = – 1

A₂₂ = (–1)²⁺² M₂₂ = M₂₂ = 3

∴ adj (A) = `[(2, -5),(-1, 3)]^"T"`

= `[(2, -1),(-5, 3)]`

∴ B = A⁻¹ = `1/|"A"|` adj (A)

∴ B = `[(2, -1),(-5, 3)]`