Is the sum or product of idempotent matrices idempotent? If you have two idempotent matrices $A$ and $B$, is $A+B$ an idempotent matrix?
Also, is $AB$ an idempotent Matrix? 
If both are true, Can I see the proof? I am completley lost in how to prove both cases.
Thanks! 
 A: If you're having difficulty proving this, that may be because it is not true (without rather restrictive additional hypotheses). Basically any pair of idempotents that is not particularly related will show this. For a very simple case:
$$
  A=\begin{pmatrix}1&0\\0&0\end{pmatrix}, \quad
  B=\begin{pmatrix}0&1\\0&1\end{pmatrix}\qquad \text{where} \quad
  A+B=\begin{pmatrix}1&1\\0&1\end{pmatrix}, \quad
  AB=\begin{pmatrix}0&1\\0&0\end{pmatrix},
$$
neither of which is idempotent.
A: Let $e_1$, $e_2 \in \mathbb{R}^n$ be linearly independent unit vectors with $c := \left\langle e_1,e_2\right\rangle \neq 0$, viewed as column vectors. For $i=1$, $2$, let $P_i := e_i e_i^T \in M_n(\mathbb{R})$ be the orthogonal projection onto $\mathbb{R}e_i$. Thus, $P_1$ and $P_2$ are idempotents with
$$
 P_1 e_1 = e_1, \quad P_1 e_2 = c e_1, \quad P_2 e_1 = c e_2, \quad P_2 e_2 = 1.
$$
Then:


*

*On the one hand,
$$
 (P_1 + P_2)e_1 = e_1 + c e_2,
$$
and on the other hand,
$$
 (P_1 + P_2)^2 e_1 = (P_1+P_2)(e_1+ce_2) = (1+c^2)e_1 + 2c e_2,
$$
so that $(P_1+P_2)^2 e_1 \neq (P_1+P_2)e_1$, and hence $(P_1+P_2)^2 = P_1+P_2$.

*On the one hand,
$$
 P_1 P_2 e_1 = P_1 (c e_2) = c^2 e_1,
$$
and on the other hand,
$$
 (P_1 P_2)^2 e_1 = P_1 P_2 (c^2 e_1) = c^4 e_1,
$$
so that since $0 < |c| < 1$, $(P_1 P_2)^2 e_1 \neq P_1 P_2 e_1$, and hence $(P_1 P_2)^2 \neq P_1 P_2$.
