Question

$If\left[\begin{array}{cc}\alpha & \beta \\ \gamma & -\alpha \end{array}\right]\text{is the square root of second order unit matrix,}Then\alpha ,\beta and\gamma \text{should satisfy the relation}$

• 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

${\alpha }^2+\beta \gamma -1=0$

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