If A, B are two idempotent matrices and AB = BA = 0 then (A + B) is

Since A and B are idempotent matrices then ${\mathrm{A}}^2=\mathrm{A}, {\mathrm{B}}^2=\mathrm{B}$ and given that AB=BA=0

 consider, ${(\mathrm{A}+\mathrm{B})}^2=(\mathrm{A}+\mathrm{B}) (\mathrm{A}+\mathrm{B})$

$$\begin{gathered} ={\mathrm{A}}^2+\mathrm{AB}+\mathrm{BA}+{\mathrm{B}}^2 \\ =\mathrm{A}+0+0+\mathrm{B} \\ \therefore {(\mathrm{A}+\mathrm{B})}^2=(\mathrm{A}+\mathrm{B}) \end{gathered}$$

$\therefore$A+B is an Idempotent matrix.

$\therefore \boxed{\mathrm{option} \left(4\right)}$