Let A be a 3 × 3 real matrix such that A 2 ( A − 2 I ) − 4 ( A − I ) = O , where I and O are the identity and null matrices, respectively. If A 5 = α A 2 + β A + γ I , where α , β and γ are real const

Question

Let A be a 3 × 3 real matrix such that A 2 ( A − 2 I ) − 4 ( A − I ) = O , where I and O are the identity and null matrices, respectively. If A 5 = α A 2 + β A + γ I , where α , β and γ are real constants, then α + β + γ is equal to:

TestHub · Accepted Answer

​ A 3 − 2 A 2 − 4 A + 4 I = 0 A 3 = 2 A 2 + 4 A − 4 I A 4 = 2 A 3 + 4 A 2 − 4 A = 2 ( 2 A 2 + 4 A − 4 I ) + 4 A 2 − 4 A A 4 = 8 A 2 + 4 A − 8 I A 5 = 8 A 3 + 4 A 2 − 8 A = 8 ( 2 A 2 + 4 A − 4 I ) + 4 A 2 − 8 A A 5 = 20 A 2 + 24 A − 32 I ∴ α = 20 , β = 24 , γ = − 32 ∴ α + β + γ = 12 ​

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