Question

$A=\left[\begin{array}{ccc}2& -2& -4\\ -1& 3& 4\\ 1& -2& -3\end{array}\right]$

Is the matrix idempotent.?

Answer 1

$A^2=A.A=\left[\begin{array}{ccc}2& -2& -4\\ -1& 3& 4\\ 1& -2& -3\end{array}\right]\times \left[\begin{array}{ccc}2& -2& -4\\ -1& 3& 4\\ 1& -2& -3\end{array}\right]$
$=\left[\begin{array}{ccc}2.2+(-2).(-1)+(-4).1& 2.(-2)+(-2).3+(-4).(-2)& 2.(-4)+(-2).4+(-4).(-3)\\ (-1).2+3.(-1)+4.1& (-1).(-2)+3.3+4.(-2)& (-1).(-4)+3.4+4.(-3)\\ 1.2+(-2).(-1)+(-3).1& 1.(-2)+(-2).3+(-3).(-2)& 1.(-4)+(-2).4+(-3).(-3)\end{array}\right]$
$=\left[\begin{array}{ccc}2& -2& -4\\ -1& 3& 4\\ 1& -2& -3\end{array}\right]=A$
Hence the matrix $A$ is idempotent.