Let $A$ be an $n^{th}$ order square matrix and matrix $B$ be its adjoint. Then $| A B + K I_{n} |$ is equal to (where $K$ is a scalar quantity)

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

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(| A | + K)^{n}

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

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

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Solution

The correct option is B $(| A | + K)^{n}$
We have, $A B = A (a d j A) = | A | I_{n}$
$∴ A B + K I_{n} = | A | I_{n} + K I_{n}$
$\Rightarrow A B + K I_{n} = (| A | + K) I_{n}$
$\Rightarrow | A B + K I_{n} | = | (| A | + K) I_{n} |$
$\Rightarrow | A B + K I_{n} | = (| A | + K)^{n} (∵ | a I_{n} | = a^{n})$

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