Solve the equation about matrix The equation is $x^2 = x$, which $x$ is a $2\times2$ matrix. Anyone can give me some hint? Thanks!
Two different hints:

1) This is a question with 2-by-2 matrices... why not just write down good ol' $x=\begin{pmatrix}a &amp; b\\c &amp; d\end{pmatrix}$, compute $x^2-x$, and see what happens?

2) You could also notice that for such a matrix $x$, we must have $x^2-x=0$; hence the minimal polynomial for $x$ must divide $t^2-t=t(t-1)$. What does that tell you about the matrix?
$x^2 - x = x(x-1)$ has single linear factors.

Consider the Jordan Normal Form. This shows that you can split the matrix up into single blocks with eigenvalues of 0 or 1.