Question

If $\left[\begin{array}{cc}\alpha & \beta \\ \gamma & -\alpha \end{array}\right]$ is to be the square root of the 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

The correct option is B ${\alpha }^2+\beta \gamma -1=0$

Given, ${\left[\begin{array}{cc}\alpha & \beta \\ \gamma & -\alpha \end{array}\right]}^2=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$
$\left[\begin{array}{cc}\alpha & \beta \\ \gamma & -\alpha \end{array}\right]\left[\begin{array}{cc}\alpha & \beta \\ \gamma & -\alpha \end{array}\right]=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$
$\left[\begin{array}{cc}{\alpha }^2+\beta \gamma & 0\\ 0& {\alpha }^2+\beta \gamma \end{array}\right]=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$
$\Rightarrow {\alpha }^2+\beta \gamma =1$
$\Rightarrow {\alpha }^2+\beta \gamma -1=0$