Let A be a matrix of order 3 such that A 2 = 3 A - 2 I where, I is an identity matrix of order 3 . If A 5 = α a + β I, then α β is equal to

Step-by-step solution

A 3 = A 2 A = 3 A - 2 I A = 3 A 2 - 2 A = 3 3 A - 2 I - 2 A = 7 A - 6 I A 4 = A 3 A = 7 A - 6 I A = 7 A 2 - 6 A = 7 3 A - 2 I - 6 A = 15 A - 14 I A 5 = A 4 A = 15 A - 14 I A = 15 A 2 - 14 A = 15 3 A - 2 I - 14 A = 31 A - 30 I ⇒ α = 31 ,   β = - 30