Question

If $\begin{bmatrix}\alpha & \beta \\ \gamma & -\alpha\end{bmatrix}$ is to be the square root of two-rowed unit matrix, then $\alpha, \beta$ and $\gamma$ should satisfy the relation

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

Answer 1

Answer: (B)

We have $\begin{bmatrix}\alpha & \beta \\ \gamma & -\alpha\end{bmatrix}\begin{bmatrix}\alpha & \beta \\ \gamma & -\alpha\end{bmatrix}=\begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}$
$\Rightarrow \begin{bmatrix}\alpha^{2}+\beta \gamma & 0 \\ 0 & \alpha^{2}+\beta \gamma\end{bmatrix}=\begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}$
$\Rightarrow \alpha^{2}+\beta \gamma-1=0$