Let A and B are square matrices of same order satisfying AB=A and BA=B, then ${(A^{2023}+B^{2023})}^{2024}$  is equal to : 

$\because A^2=A.A=ABA=AB=A$

$\begin{array}{l}and{B}^2=B.AB=BA=B\\ A=A^2=A^3=\mathrm{....}andB=B^2=B^3=\mathrm{........}\\ {(A^{2023}+B^{2023})}^{2024}={(A+B)}^{2024}\\ {(A+B)}^2=A^2+B^2+AB+BA=2(A+B)\\ {(A+B)}^3=2(A+B)(A+B)=2^2(A+B)\\ {(A+B)}^{2024}=2^{2023}(A+B)\end{array}$