If $A$ and $B$ are two square matrices of same order such that $AB = A$ and $BA = B$, then the value of $A^{2024} + B^{2024}$ is equal to

$A + B$

The correct answer is Option (3) → $A + B$

Given: AB = A and BA = B

From AB = A ⇒ A(B − I) = 0

From BA = B ⇒ (A − I)B = 0

Multiply AB = A on left by A:

A²B = A² ⇒ using AB = A ⇒ A² = A

Similarly, multiply BA = B on left by B:

B²A = B² ⇒ using BA = B ⇒ B² = B

Hence, both A and B are idempotent matrices:

$A^n = A$ and $B^n = B$ for all $n \ge 1$

Therefore,

$A^{2024} + B^{2024} = A + B$

$A^{2024} + B^{2024} = A + B$