Let A be a matrix of order \(2 \times 3\) and B a matrix of order \(3 \times 2\). Which of the following statements are correct?
(A) AB is a square matrix of order 2.
(B) BA is a square matrix of order 3.
(C) \(A + B'\) is defined.
(D) \(A + B\) is defined.
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (A), (B) and (C) only
A product PQ is defined when the number of columns of P equals the number of rows of Q, and then PQ has the rows of P and the columns of Q. A sum is defined only for matrices of the same order.
\(AB:\ (2 \times 3)(3 \times 2) = 2 \times 2\)
\(BA:\ (3 \times 2)(2 \times 3) = 3 \times 3\)
\(B'\) is of order \(2 \times 3\)
So AB is of order \(2 \times 2\), BA is of order \(3 \times 3\), \(B'\) has the same order as A, and B does not.
Checking the statements
(A) AB is a square matrix of order 2 — Correct. \((2 \times 3)(3 \times 2)\) gives \(2 \times 2\).
(B) BA is a square matrix of order 3 — Correct. \((3 \times 2)(2 \times 3)\) gives \(3 \times 3\).
(C) \(A + B'\) is defined — Correct. Both A and \(B'\) are of order \(2 \times 3\).
(D) \(A + B\) is defined — Incorrect. A is \(2 \times 3\) but B is \(3 \times 2\).