Let A be a square matrix of order n and B be its adjoint, then for a scalar K, $| A B + K I_{n} |$ is ___

$(| A | + K)^{n - 2}$

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$(| A | + K)^{n}$

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$(| A | + K)^{n - 1}$

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None of these

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Solution

The correct option is B
$(| A | + K)^{n}$

We have AB = A(adj A) $= | A | I_{n}$
$∴ A B + K I_{n} = | A | I_{n} + K I_{n} \Rightarrow A B + K I_{n} = (| A | + K) I_{n} \Rightarrow | A B + K I_{n} | = | (| A | + K) I_{n} | = (| A | + K)^{n}$

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