Question

Find the inverse of the following matrices by the adjoint method `[(2, -2),(4, 5)]`.

Answer 1

Let A = `[(2, -2),(4, 5)]`

∴ |A| = `[(2, -2),(4, 5)]` = 10 + 8 = 18 ≠ 0

∴ A^–1 exists.
A₁₁ = (– 1)¹⁺¹ M₁₁ = (1)(5) = 5
A₁₂ = (– 1)¹⁺² M₁₂ = (– 1)(4) = – 4
A₂₁ = (– 1)²⁺¹ M₂₁ = (– 1)(– 2) = 2
A₂₂ = (– 1)²⁺² M₂₂ = (1)(2) = 2
∴  The matrix of the co-factors is

[A_ij]_2x2 = `[("A"_11, "A"_12),("A"_21, "A"_22)] = [(5, -4),(2, 2)]`

Now adj A = `["A"_"ij"]_(2xx2)^"T" = [(5, 2),(-4, 2)]`

∴ A–1 = `(1)/|"A"|("adj A")`

= `(1)/(18)[(5, 2),(-4, 2)]`.