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Transpose of a Matrix

The number of...

Question

The number of diagonal matrix $A$ of order $n$ for which $A^{3} = A$ is

1

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0

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2

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3

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Solution

The correct option is D $3$
let the diagonal matrix $A = D i a g [d, d, d, \dots n t i m e s]$
$\Rightarrow A^{3} = D i a g [d^{3}, d^{3}, d^{3}, \dots n t i m e s]$
but $A^{3} = A$
$\Rightarrow d^{3} = d$
$∴ d = - 1, 0, 1$
i.e $3$ matrices are possible.
Hence, option D.

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