Question

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

• A. $ 1 + \alpha^2 + \beta \gamma = 0$
• B. $ 1 + \alpha^2 - \beta \gamma = 0$
• C. $ 1 - \alpha^2 + \beta \gamma = 0$
• D. $\alpha^2 + \beta \gamma = 1$

Answer 1

The Correct Option is D

$\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 $