Question

If matrix $A=\left[\begin{array}{cc}0& -a\\ -b& 0\end{array}\right]$ is square root of $\left[\begin{array}{cc}3& 0\\ 0& 3\end{array}\right],$ then the number of such possible matrix
is(are)
(where $a,b\in N$)

Answer 1

Given : $A^2=\left[\begin{array}{cc}3& 0\\ 0& 3\end{array}\right]$
$\Rightarrow \left[\begin{array}{cc}0& -a\\ -b& 0\end{array}\right]\left[\begin{array}{cc}0& -a\\ -b& 0\end{array}\right]=\left[\begin{array}{cc}3& 0\\ 0& 3\end{array}\right]\Rightarrow \left[\begin{array}{cc}ab& 0\\ 0& ab\end{array}\right]=\left[\begin{array}{cc}3& 0\\ 0& 3\end{array}\right]$
$\Rightarrow ab=3$
$\Rightarrow a=1,b=3$ or $a=3,b=1$
So, two such matrices are possible.