Question

If matrix $A=\left[\begin{array}{cc}a& 0\\ 0& b\end{array}\right]$ is square root of $\left[\begin{array}{cc}4& 0\\ 0& 9\end{array}\right],$ then the number of such possible matrices is
(where $a,b\in R$)

Answer 1

As, matrix $A$ is square root of $\left[\begin{array}{cc}4& 0\\ 0& 9\end{array}\right]$
$\Rightarrow A^2=\left[\begin{array}{cc}4& 0\\ 0& 9\end{array}\right]\cdots \left(i\right)$ and
$A^2=\left[\begin{array}{cc}a& 0\\ 0& b\end{array}\right]\left[\begin{array}{cc}a& 0\\ 0& b\end{array}\right]=\left[\begin{array}{cc}{a}^2& 0\\ 0& b^2\end{array}\right]$ comparing with $\left(i\right)$
we get, $a=\pm 2,b=\pm 3$
So, there are $4$ such matrices possible.