Question

If A = `[(1, 3, 3),(3, 1, 3),(3, 3, 1)]`, then show that A² – 5A is a scalar matrix

Answer 1

A² – 5A = A.A – 5A

= `[(1, 3, 3),(3, 1, 3),(3, 3, 1)] [(1, 3, 3),(3, 1, 3),(3, 3, 1)] - 5[(1, 3, 3),(3, 1, 3),(3, 3, 1)]`

= `[(1 + 9 + 9, 3 + 3 + 9, 3 + 9 + 3),(3 + 3 + 9, 9 + 1 + 9, 9 + 3 + 3),(3 + 9 + 3, 9 + 3 + 3, 9 + 9 + 1)] - [(5, 15, 15),(15, 5, 15),(15, 5, 15)]`

= `[(19, 5, 15),(15, 19, 15),(15, 5, 9)] - [(5, 15, 15),(15, 5, 15),(15, 15, 5)]`

= `[(19 - 5, 15 - 15, 15 - 15),(15 - 15, 19 - 5, 15 - 15),(15 - 15, 15 - 15, 19- 5)]`

= `[(14, 0, 0),(0, 14, 0),(0, 0, 14)]`, which is a scalar matrix.