Question

Find the inverse of the following matrix by the adjoint method:

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

Answer 1

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

∴ |A| = `|(3, - 1),(2, -1)|`

|A| = – 3 + 2

|A| = – 1

|A| ≠ 0

∴ A^–1 exists.

A₁₁ = (– 1)¹⁺¹ M₁₁ = (1)(–1) = – 1

A₁₂ = (– 1)¹⁺² M₁₂ = (– 1)(2) = – 2

A₂₁ = (– 1)²⁺¹ M₂₁ = (– 1)(– 1) = 1

A₂₂ = (– 1)²⁺² M₂₂ = (1)(3) = 3

∴  The matrix of the co-factors is

[A_ij]_2×2 = `[("A"_11, "A"_12),("A"_21, "A"_22)]`

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

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

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

∴ A^–1 = `(1)/|"A"|("adj A")`

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

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