If A and B are two matrices such that both \(A + B\) and \(AB\) are defined, then
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → A and B are square matrices of the same order
Let A be of order \(m \times n\).
\(A + B\) is defined, so B is also of order \(m \times n\) (same order).
\(AB\) is defined, so the n columns of A must match the m rows of B: \(n = m\).
Hence A and B are both of order \(m \times m\), that is, square matrices of the same order.
Why the other options are incorrect
Option (1) A and B are of orders \(m \times n\) and \(n \times m\) respectively, with \(m \ne n\) — Incorrect. Matrices of different orders cannot be added.
Option (2) A is a row matrix and B is a column matrix, each having 3 elements — Incorrect. A is \(1 \times 3\) and B is \(3 \times 1\), so \(A + B\) is not defined.
Option (3) A and B are of the same order \(m \times n\), with \(m \ne n\) — Incorrect. Then AB needs \(n = m\), so AB is not defined.