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  4. If [α&β γ&-α] is square root of identity matrix of order 2 then

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Q. If $\begin{bmatrix}\alpha&\beta\\ \gamma&-\alpha\end{bmatrix}$ is square root of identity matrix of order $2$ then

BITSATBITSAT 2014

Solution:

$\begin{bmatrix}\alpha&\beta\\ \gamma&-\alpha\end{bmatrix} = \sqrt{I_{2}} ; $
$ \begin{bmatrix}\alpha&\beta\\ \gamma&-\alpha\end{bmatrix} \begin{bmatrix}\alpha &\beta \\ \gamma &-\alpha \end{bmatrix} = \begin{bmatrix}1&0\\ 0&1\end{bmatrix} $
$\Rightarrow \alpha^{2} + \beta\gamma = 1 $