If A,B are two square matrices such that $AB=A,BA=B$  then ${\left(A+B\right)}^7=$

$AB=A,BA=B\Rightarrow A^2=A,B^2=B$

$\Rightarrow {\left(A+B\right)}^2=A^2+AB+BA+B^2=A+A+B+B=2^1\left(A+B\right)$

$\Rightarrow$  ${\left(A+B\right)}^3={\left(A+B\right)}^2\left(A+B\right)=\left(2\left(A+B\right)\right)\left(A+B\right)=2^2\left(A+B\right)$

$\Rightarrow$  ${\left(A+B\right)}^n=2^{n-1}\left(A+B\right)$