Question

Find the inverse of the matrix A by using adjoint method.

where A = `[(-3, -1, 1),(0, 0, 1),(-15, 6, -6)]`

Answer 1

A = `[(-3, -1, 1),(0, 0, 1),(-15, 6, -6)]`

∴ |A| = – 3(0 – 6) + 1(0 + 15) + 1(0 – 0)

= 18 + 15

= 33 ≠ 0

∴ A^–1 exists

To find cofactors.

m₁₁ = 0 – 6 = – 6

m₁₂ = 0 + 15 = 15

m₁₃ = 0 + 0 = 0

m₂₁ = 6 – 6 = 0

m₂₂ = 18 + 15 = 33

m₂₃ = – 18 – 15 = – 33

m₃₁ = – 1 – 0 = – 1

m₃₂ = – 3 – 0 = – 3

m₃₃ = 0 – 0 = 0

A₁₁ = (– 1)² (– 6) = – 6

A₁₂ = (– 1)³.15 =  – 15

A₁₃ = (– 1)⁴.0 = 0

A₂₁ = (– 1)³.0 = 0

A₂₂ = (– 1)⁴.33 = 33

A₂₃ = (– 1)⁵. (– 33) = 33

A₃₁ = (– 1)⁴ (– 1) = – 1

A₃₂ = (– 1)⁵ (– 3) = 3

A₃₃ = (– 1)⁶.0 = 0

∴ Matrix of co-factors

= `[(-6, -15, 0),(0, 33, 33),(-1, 3, 0)]`

adj.A = `[(-6, 0, -1),(-15, 33, 3),(0, 33, 0)]`

A^–1 = `1/|A|` (adj A)

∴ A^–1 = `1/33 [(-6, 0, -1),(-15, 33, 3),(0, 33, 0)]`