Question

The sum of two idempotent matrices $A$ and $B$ is idempotent if $AB=BA=$ .......................

Answer 1

Given A and B are idempotent matrices
$A^2=A$ and $B^2=B$
Consider, $(A+B{)}^2=(A+B\left)\right(A+B)$
$=A^2+AB+BA+B^2$
$=A+B+AB+BA$
For, $(A+B{)}^2=A+B$ to hold, $AB=BA=0$