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Adjoint of a Matrix

If n is eve...

Question

If $n$ is even and $A^{n} = A^{- 1}$ , then $($ adj $A)^{- 1} =$

A

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A^{2}

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A^{T}

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does not exist

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Solution

The correct option is A $A$
Given $A^{n} = A^{- 1}$
$\Rightarrow A^{n + 1} = A A^{- 1}$
$\Rightarrow A^{n + 1} = I$
Put $n = 0$
$A = I$
$| A | = 1$
Now, $(a d j A)^{- 1} = \frac{a d j (a d j A)}{| a d j A |}$
$= \frac{| A |^{n - 2} A}{| A |^{n - 1}}$
$= \frac{A}{| A |}$
$\Rightarrow (a d j A)^{- 1} = A$ -------(Since $| A | = 1$ )

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