Question

Let X be an unknown matrix of order 2 and I be an identity matrix of order 2, then the
equation $X^2=1$

• A. is satisfied if $X=I$
• B. is satisfied if $X=\left[\begin{array}{cc}7i& 10\\ 5& -7i\end{array}\right]$
• C. is satisfied if $X=\left[\begin{array}{cc}\pm \sqrt{1-\lambda \mu }& \lambda \\ \mu & \pm \sqrt{1-\lambda \mu }\end{array}\right]$
• D. has no solution except identity matrix

Answer 1

Answer: (A), (B)

is satisfied if $X=I$
is satisfied if $X=\left[\begin{array}{cc}7i& 10\\ 5& -7i\end{array}\right]$

$I=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$
$X^2=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]=1$
$X^2=\left[\begin{array}{cc}7i& 10\\ 5& -7i\end{array}\right]\left[\begin{array}{cc}7i& 10\\ 5& -7i\end{array}\right]=1$
Similarly,
$X=\left[\begin{array}{cc}\pm \sqrt{1-\lambda \mu }& \lambda \\ \mu & \pm \sqrt{1-\lambda \mu }\end{array}\right]$ is not a solution of the given equation.
Hence, options 'A', 'B' are correct.