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.

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