Question

Let A = [( 2 3 ),( –1 2)]. Then show that A² – 4A + 7I = O. Using this result calculate A⁵ also.

Answer 1

We have,

Therefore, A² – 4A + 7I

A² = 4A – 7I

Thus A³ = A.A² = A (4A – 7I) = 4 (4A – 7I) – 7A

= 16A – 28I – 7A = 9A – 28I

and so A⁵ = A³A²

= (9A – 28I) (4A – 7I)

= 36A² – 63A – 112A + 196I

= 36 (4A – 7I) – 175A + 196I

= – 31A – 56I