Find all $n$ by $n$ matrices $A$ such that $A^2 = A.$

Question

It isn't too hard to show that there are infinitely many non-invertible matrices, and finitely many invertible ones. But how do we find all the possible matrices?

Edit : The other question has not given all the matrices.

Answer 1

Notice that $A$ is diagonalizable since the polynomial $x(x-1)$ annihilates it and then $\operatorname{sp}(A)\subset \{0,1\}$ with multiplicity $\dim\ker A$ and $\operatorname{rank}A=:r$ respectively so $A$ is similar to $D=\operatorname{diag}(\underbrace{1,\ldots,1}_{r\,\text{times}},0,\ldots,0)$, i.e.

$$A=PDP^{-1}$$ where $P$ is invertible.

Reciprocally, the matrix $A=PDP^{-1}$ verify $A^2=A$. Conclude.