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
Answer & explanation
Correct answer: option 3
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,
$A^{2024} + B^{2024} = A + B$
$A^{2024} + B^{2024} = A + B$