Find 2 idempotent matrix with some poperties.

Question

I want to find an example of two idempotent matrices $a,b$ with entries in $\mathbb{Z}_2$ with $a+b$ also idempotent, $ab\not=0$ and $a\not=b$. Can someone find one?

I have prove that if you work in a field with characteristic greater than 2 this is not possible. But I have been asked to give this example in $\mathbb{Z}_2$.

Answer 1

Try: $$ a = \left(\matrix{1&0&0\\0&1&0\\0&0&0}\right) \quad b = \left(\matrix{0&0&0\\0&1&0\\0&0&1}\right) $$