For a 3 × 3 matrix M , let trace ( M ) denote the sum of all the diagonal elements of M . Let A be a 3 × 3 matrix such that ∣ A ∣ = 2 1 and trace ( A ) = 3 . If B = adj ( adj ( 2 A )) , then the val

Question

For a 3 × 3 matrix M , let trace ( M ) denote the sum of all the diagonal elements of M . Let A be a 3 × 3 matrix such that ∣ A ∣ = 2 1 ​ and trace ( A ) = 3 . If B = adj ( adj ( 2 A )) , then the value of ∣ B ∣ + trace (B) equals :

TestHub · Accepted Answer

∵ tr ( A ) = 3 and ∣ A ∣ = 2 1 ​ Now, B = adj ( adj ( 2 A )) = ∣2 A ∣ 3 − 2 ⋅ ( 2 A ) = 2 3 ∣ A ∣ ⋅ 2 A = 8 A ​ ∴ tr ( B ) = 8 tr ( A ) = 24 and ∣ B ∣ = ∣8 A ∣ = 8 3 ⋅ 2 1 ​ = 256 ∴ trace ( B ) + ∣ B ∣ = 24 + 256 ​ = 280

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