Question

If $\left[\begin{array}{cc}\alpha & \beta \\ \gamma & -\alpha \end{array}\right]$ to the square is two rowed unit matrix, then $\alpha ,\beta ,\gamma$ should satisfy the relation

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

Answer 1

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

since $\left[\begin{array}{cc}\alpha & \beta \\ \gamma & -\alpha \end{array}\right]$ is a square root of $I_2$ i.e., two rowed unit matrix
$\therefore {\left[\begin{array}{cc}\alpha & \beta \\ \gamma & -\alpha \end{array}\right]}^2=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$
$\Rightarrow \left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]=\left[\begin{array}{cc}\alpha & \beta \\ \gamma & -\alpha \end{array}\right]\left[\begin{array}{cc}\alpha & \beta \\ \gamma & -\alpha \end{array}\right]$
$\Rightarrow \left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]=\left[\begin{array}{cc}{\alpha }^2+\beta \gamma & 0\\ 0& {\alpha }^2+\beta \gamma \end{array}\right]$
$\therefore {\alpha }^2+\beta \gamma =1$
$\Rightarrow 1-{\alpha }^2-\beta \gamma =0$