Let A be an nth-order square matrix and B be its adjoint, then $| A B + K I_{n} |$ is (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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none of these

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Solution

The correct option is B $(| A | + K)^{n}$
Given $B = a d j A$
$\Rightarrow | B | = | a d j A |$
$\Rightarrow | B | = | A |^{n - 1}$
Now, $| A B + K I_{n} | = | A B | + | K I_{n} |$
$= | A | | B | + K^{n} | I |$
$= | A |^{n} + K^{n}$
$= (| A | + K)^{n}$
$\Rightarrow | A B + K I_{n} | = (| A | + K)^{n}$

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