If A and B are square matrices of order n, such that AB = A and BA = B, then A2 =
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → A
Since A = AB and BA = B, and matrix multiplication is associative:
\( \displaystyle A^2=A\cdot A=(AB)A \)
\( \displaystyle =A(BA)=AB \)
\( \displaystyle =A \)
Why the other options are incorrect
Option (2) B — Incorrect. A2 = A, and A need not equal B; for example \( A=\begin{bmatrix}1&0\\0&0\end{bmatrix} \) and \( B=\begin{bmatrix}1&0\\1&0\end{bmatrix} \) satisfy AB = A and BA = B, but A ≠ B.
Option (3) B2 — Incorrect. In the same way \( B^2=(BA)B=B(AB)=BA=B \), so B2 = B, which need not equal A.
Option (4) B2A — Incorrect. \( B^2A=BA=B \), which again need not equal A.