Let A = 2 1 0 a 3 5 0 1 b . If A 3 = 4 A 2 − A − 21 I , where I is the identity matrix of order 3 × 3 , then 2 a + 3 b is equal to

Question

Let A = ​ 2 1 0 ​ a 3 5 ​ 0 1 b ​ ​ . If A 3 = 4 A 2 − A − 21 I , where I is the identity matrix of order 3 × 3 , then 2 a + 3 b is equal to

TestHub · Accepted Answer

​ A 3 − 4 A 2 + A + 21 I = 0 tr ( A ) = 4 = 5 + 6 ⇒ b = − 1 ∣ A ∣ = − 21 − 16 + a = − 21 ⇒ a = − 5 2 a + 3 b = − 13 ​

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