Question

If A = `[(1, 2, 2),(2, 1, 2),(2, 2, 1)]`, show that A² – 4A is a scalar matrix.

Answer 1

A2 – 4A = A.A – 4A
= `[(1, 2, 2),(2, 1, 2),(2, 2, 1)][(1, 2, 2),(2, 1, 2),(2, 2, 1)] - 4[(1, 2, 2),(2, 1, 2),(2, 2, 1)]`

= `[(1 + 4 + 4, 2 + 2 + 4, 2 + 4 + 2),(2 + 2 + 4, 4 + 1 + 4, 4+ 2 + 2),(2 + 4 + 2, 4 + 2 + 2, 4 + 4 + 1)] - [(4, 8, 8),(8, 4, 8),(8, 8, 4)]`

= `[(9, 8, 8),(8, 9, 8),(8, 8, 9)] - [(4, 8, 8),(8, 4, 8),(8, 8, 4)]`

= `[(9 - 4, 8 - 8, 8 - 8),(8 - 8, 9 - 4, 8 - 8),(8 - 8, 8 - 8, 9 - 4)]`

= `[(5, 0, 0),(0, 5, 0),(0, 0, 5)]`, which is a scalar martix.