Let A be a matrix of order \(3 \times 4\) and B a matrix of order \(4 \times 3\). Consider the following statements:
(A) Both AB and BA are defined.
(B) AB = BA is possible for suitable A and B.
(C) \(A + B'\) is defined.
(D) \(A'B'\) is of order \(3 \times 3\).
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (A) and (C) only
A product PQ of a \(m \times n\) matrix P and a \(n \times p\) matrix Q is defined and has order \(m \times p\); a sum needs both matrices of the same order.
AB: \((3 \times 4)(4 \times 3)\) gives \(3 \times 3\)
BA: \((4 \times 3)(3 \times 4)\) gives \(4 \times 4\)
B' is of order \(3 \times 4\), A' is of order \(4 \times 3\)
A'B': \((4 \times 3)(3 \times 4)\) gives \(4 \times 4\)
So AB is \(3 \times 3\), BA is \(4 \times 4\), B' has the same order \(3 \times 4\) as A, and A'B' is \(4 \times 4\).
Checking the statements
(A) Both AB and BA are defined. — Correct. AB is \(3 \times 3\) and BA is \(4 \times 4\).
(B) AB = BA is possible for suitable A and B. — Incorrect. A \(3 \times 3\) matrix can never equal a \(4 \times 4\) matrix.
(C) \(A + B'\) is defined. — Correct. A and B' are both of order \(3 \times 4\).
(D) \(A'B'\) is of order \(3 \times 3\). — Incorrect. \((4 \times 3)(3 \times 4)\) gives order \(4 \times 4\).