Question

If A = `[(4, 3, 2),(-1, 2, 0)]`, B = `[(1, 2),(-1, 0),(1, -2)]`

Find (AB)^–1 by adjoint method.

Solution:

AB = `[(4, 3, 2),(-1, 2, 0)] [(1, 2),(-1, 0),(1, -2)]`

AB = [  ]

|AB| =  `square`

M₁₁ = –2  ∴ A₁₁ = (–1)¹⁺¹ . (–2) = –2

M₁₂ = –3     A₁₂ = (–1)¹⁺² . (–3) = 3

M₂₁ = 4       A₂₁ = (–1)²⁺¹ . (4) = –4

M₂₂ = 3       A₂₂ = (–1)²⁺² . (3) = 3

Cofactor Matrix [Aij] = `[(-2, 3),(-4, 3)]`

adj (A) = [  ]

A^–1 = `1/|A| . adj(A)`

A^–1 = `square`

Answer 1

AB = `[(4, 3, 2),(-1, 2, 0)] [(1, 2),(-1, 0),(1, -2)]`

AB = `bb([(3, 4),(-3, -2)])`

|AB| = 6

M₁₁ = –2  ∴ A₁₁ = (–1)¹⁺¹ . (–2) = –2

M₁₂ = –3     A₁₂ = (–1)¹⁺² . (–3) = 3

M₂₁ = 4       A₂₁ = (–1)²⁺¹ . (4) = –4

M₂₂ = 3       A₂₂ = (–1)²⁺² . (3) = 3

Cofactor Matrix [Aij] = `[(-2, 3),(-4, 3)]`

adj (A) = `bb([(-2, -4),(3, 3)])`

A^–1 = `1/|A| . adj(A)`

A^–1 = `1/6*bb([(-2, -4),(3, 3)])`