\[A\]
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\[\begin{equation} \begin{aligned} \left( \begin{array}{cc} 25 & -20 \\ 30 & -24 \end{array} \right)\left( \begin{array}{cc} 25 & -20 \\ 30 & -24 \end{array} \right) &= \left( \begin{array}{cc} 25 \times 25-20 \times 30 & 25 \times-20-20 \times -24 \\ 30 \times 25 -24 \times 30 & 30 \times -20 -24 \times -24 \end{array} \right) \\ &= \left( \begin{array}{cc} 25 & \ -20 \\ 30 & -24 \end{array} \right) \end{aligned} \end{equation}\]
The identity and trivial or zero matrices are also idempotent.