Question

If $A=\left[\begin{array}{cc}4& -1\\ x& -3\end{array}\right]$ is an idempotent matrix, then the value of $x$ is

Answer 1

Given $A=\left[\begin{array}{cc}4& -1\\ x& -3\end{array}\right]$ is an idempotent matrix.
We know that for an idempotent matrix, $A^2=A$
$A^2=\left[\begin{array}{cc}4& -1\\ x& -3\end{array}\right]\left[\begin{array}{cc}4& -1\\ x& -3\end{array}\right]$
$=\left[\begin{array}{cc}16-x& -1\\ x& 9-x\end{array}\right]=\left[\begin{array}{cc}4& -1\\ x& -3\end{array}\right]$
Equating the terms, we get
$16-x=4\Rightarrow x=12$
or,
$9-x=-3\Rightarrow x=12$
$\therefore x=12$