## Idempotent Matrices

A matrix
$A$
is idempotent if
$A^2=P$
. This means that only a square matrix can be idempotent.
Example:
$A= \left( \begin{array}{cc} \25 & \ -20 \\ 30 & -24 \end{array} \right)$
is idemmpotent, since
$\left( \begin{array}{cc} \25 & \ -20 \\ 30 & -24 \end{array} \right)\left( \begin{array}{cc} \25 & \ -20 \\ 30 & -24 \end{array}= \right)\left( \begin{array}{cc} \25 \times 25-20 \times 30 & \ 25 \times-20-20 \times -24 \\ 30 \times 25 -24 \times 30 & 30 \times -20 -24 \times -24 \end{array} \right) = \left( \begin{array}{cc} \25 & \ -20 \\ 30 & -24 \end{array} \right)$

The identity and trivial or zero matrices are also idempotent.