Q. 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) $1 - \alpha^{2} - \beta\gamma = 0$
Explanation
(b)
We have,
$\begin{bmatrix}
\alpha & \beta \\
\gamma & - \alpha\
\end{bmatrix}\begin{bmatrix}
\alpha & \beta \\
\gamma & - \alpha\
\end{bmatrix} = \begin{bmatrix}
1 & 0 \\
0 & 1\
\end{bmatrix}$
$\Rightarrow \begin{bmatrix}
\alpha^{2} + \beta\gamma & 0 \\
0 & \alpha^{2} + \beta\gamma\
\end{bmatrix}$=$\begin{bmatrix}
1 & 0 \\
0 & 1\
\end{bmatrix}$
$\Rightarrow \alpha^{2} + \beta\gamma - 1 = 0$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.