Question

If $A=\left[\begin{array}{ccc}2& -2& -4\\ -1& 3& 4\\ 1& -2& x\end{array}\right]$ is an idempotent matrix then find $|x|=$ .....................

Answer 1

Given $A=\left[\begin{array}{ccc}2& -2& -4\\ -1& 3& 4\\ 1& -2& x\end{array}\right]$ is an idempotent matrix.
$\Rightarrow A^2=A$
$\Rightarrow \left[\begin{array}{ccc}2& -2& -4\\ -1& 3& 4\\ 1& -2& x\end{array}\right]\left[\begin{array}{ccc}2& -2& -4\\ -1& 3& 4\\ 1& -2& x\end{array}\right]=\left[\begin{array}{ccc}2& -2& -4\\ -1& 3& 4\\ 1& -2& x\end{array}\right]$
$\Rightarrow \left[\begin{array}{ccc}2& -2& -4x-16\\ -1& 3& 4x+16\\ x+4& -2x-8& x^2-12\end{array}\right]=\left[\begin{array}{ccc}2& -2& -4\\ -1& 3& 4\\ 1& -2& x\end{array}\right]$
$\Rightarrow x=-3$
$\therefore |x|=3$