Question

What is an idempotent matrix?

Answer 1

$An\ idempotent\ matrix\ is\ a\ matrix\ n\times n\ (square\ matrix\ A)\ which:A^2=A\\\\Example:\\ A=\left[\begin{array}{ccc}2&-1\\2&-1\end{array}\right]\\A^2=\left[\begin{array}{ccc}2&-1\\2&-1\end{array}\right]\cdot\left[\begin{array}{ccc}2&-1\\2&-1\end{array}\right]=\left[\begin{array}{ccc}2\cdot2-1\cdot2&2\cdot(-1)-1\cdot(-1)\\2\cdot2-1\cdot2&2\cdot(-1)-1\cdot(-1)\end{array}\right]\\= \left[\begin{array}{ccc}4-2&-2+1\\4-2&-2+1\end{array}\right] = \left[\begin{array}{ccc}2&-1\\2&-1\end{array}\right] =A$

Answer 2

A matrix E satisfying the equation E2 = E.