Question

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

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

Answer 1

Answer: (B)

We have,
$\left[\begin{array}{cc}\alpha & \beta \\ \gamma & -\alpha \end{array}\right]={\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]}^{1/2}$
$\Rightarrow {\left[\begin{array}{cc}\alpha & \beta \\ \gamma & -\alpha \end{array}\right]}^2=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$
[Taking square both sides ]
$\Rightarrow \left[\begin{array}{cc}{\alpha }^2+\beta \gamma & \alpha \beta -\alpha \beta \\ \alpha \gamma -\alpha \gamma & \beta \gamma +{\alpha }^2\end{array}\right]=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$
$\Rightarrow \left[\begin{array}{cc}{\alpha }^2+\beta \gamma & 0\\ 0& {\alpha }^2+\beta \gamma \end{array}\right]=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$
On comparing both sides, we get
${\alpha }^2+\beta \gamma =1$
$\Rightarrow 1-{\alpha }^2-\beta \gamma =0$