Suppose A is any 3 × 3 non-singular matrix and ( A − 3 I ) ( A − 5 I ) = O , where I = I 3 and O = O 3 . If α A + β A − 1 = 4 I , then α + β is equal to

Question

Suppose A is any 3 × 3 non-singular matrix and ( A − 3 I ) ( A − 5 I ) = O , where I = I 3 ​ and O = O 3 ​ . If α A + β A − 1 = 4 I , then α + β is equal to

TestHub · Accepted Answer

We have ​ ( A − 3 I ) ( A − 5 I ) = O ⇒ A 2 − 8 A + 15 I = O ​ Multiplying both sides by A − 1 , we get; A − 1 A ⋅ A − 8 A − 1 A + 15 A − 1 I = A − 1 O ⇒ ∴ ​ A − 8 I + 15 A − 1 = O A + 15 A − 1 = 8 I 2 A ​ + 2 15 A − 1 ​ = 4 I α + β = 2 1 ​ + 2 15 ​ = 2 16 ​ = 8 ​

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