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) → \(A + B\)
Using \(AB = A\) and \(BA = B\) in turn, with the associative law:
\(A^2 = A\cdot A = (AB)A\)
\(= A(BA) = AB = A\)
In the same way,
\(B^2 = B\cdot B = (BA)B\)
\(= B(AB) = BA = B\)
So A and B are idempotent. Then \(A^3 = A^2\cdot A = A\cdot A = A\), and so on: \(A^n = A\) and \(B^n = B\) for every \(n \ge 1\).
\(A^{2024} + B^{2024} = A + B\)