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Adjoint of a Matrix

If A is a squ...

Question

If A is a square matrix so that $A a d j A = d i a g (k, k, k)$ then $| a d j A | =$

k

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k^{2}

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k^{3}

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k^{6}

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Solution

The correct option is D $k^{6}$
Given, $A a d j A = d i a g (k, k, k)$
$\Rightarrow | A | I_{n} = k^{3} (∵ A a d j A = | A | I_{n})$
$\Rightarrow | A | = k^{3}$ and $n = 3$
We know $| a d j A | = | A |^{n - 1}$
Here, $| a d j A | = (k^{3})^{2} = k^{6}$

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-

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A (a d j A) = | A |

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\frac{| A d j (A d j A)}{| A d j A |}

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A = ⎡ ⎢ ⎢ ⎣ \begin{matrix} 2 k - 1 & 2 \sqrt{k} & 2 \sqrt{k} 2 \sqrt{k} & 1 & - 2 k - 2 \sqrt{k} & 2 k & - 1 \end{matrix} ⎤ ⎥ ⎥ ⎦

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(a d j A) + d e t (a d j B) = 10^{6}, t h e n [k]

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