Question

If square root of the matrix $A=\left[\begin{array}{cc}2& 0\\ 0& 0\end{array}\right]$ is in the form of $\left[\begin{array}{cc}a& 0\\ 0& b\end{array}\right],$ then number of possible matrices is

Answer 1

Given : $A=\left[\begin{array}{cc}2& 0\\ 0& 0\end{array}\right]$
$\sqrt{A}=\left[\begin{array}{cc}a& 0\\ 0& b\end{array}\right]$
$\Rightarrow \left[\begin{array}{cc}a& 0\\ 0& b\end{array}\right]\left[\begin{array}{cc}a& 0\\ 0& b\end{array}\right]=\left[\begin{array}{cc}2& 0\\ 0& 0\end{array}\right]$
$\Rightarrow \left[\begin{array}{cc}{a}^2& 0\\ 0& b^2\end{array}\right]=\left[\begin{array}{cc}2& 0\\ 0& 0\end{array}\right]$
$a^2=2$ and $b^2=0$
So, we have two possible pairs of $a$ and $b$
$(\sqrt{2},0),(-\sqrt{2},0)$
$\therefore$ Two matrices possible.
i.e., $\left[\begin{array}{cc}\sqrt{2}& 0\\ 0& 0\end{array}\right],\left[\begin{array}{cc}-\sqrt{2}& 0\\ 0& 0\end{array}\right]$