Question

If A = `[(0, 4, 3),(1, -3, -3),(-1, 4, 4)]`, then find A² and hence find A⁻¹ 

Answer 1

|A| = `0 - 4|(1, -3),(-1, 4)| + 3|(1, -3),(-1, 4)|`

= – 4(4 – 3) + 3(4 – 3)

= –1 ≠ 0

∴ A⁻¹ exist

A² = `[(0, 4, 3),(1, -3, -3),(-1, 4, 4)] [(0, 4, 3),(1, -3, -3),(-1, 4, 4)]`

= `[(0 + 4 - 3, 0 - 12 + 12, 0 - 12 + 12),(0 - 3 + 3, 4 + 9 - 12, 3 + 9 - 12),(0 + 4 - 4, -4 - 12 + 16, -3 - 12 + 16)]`

= `[(1, 0, 0),(0, 1, 0),(0, 0, 1)]`

i.e., A × A = I

∴ A^–1 × A × A = A^–1 × I

∴ A = A^–1 

∴ A^–1 = `[(0, 4, 3),(1, -3, -3),(-1, 4, 4)]`