[Math] Is the sum or product of idempotent matrices idempotent

linear algebramatrices

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!

Best Answer

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.

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