If \(A=\begin{bmatrix}2 & 1 & 3\\4 & -3 & 5\end{bmatrix}\) and \(B=\begin{bmatrix}-2 & 3\\4 & -5\\1 & 2\end{bmatrix}\), then which of the following statements are TRUE?
(A) AB is defined
(B) AB and BA both are defined and AB = I, where I is an identity matrix of order 2
(C) BA is defined
(D) AB and BA both are defined and AB = BA
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (A) and (C) only
A is of order \(2 \times 3\) and B is of order \(3 \times 2\). A product PQ is defined when the number of columns of P equals the number of rows of Q.
Checking the statements
(A) AB is defined — Correct. A has 3 columns and B has 3 rows, so AB is defined, of order \(2 \times 2\).
(B) AB and BA both are defined and AB = I, where I is an identity matrix of order 2 — Incorrect. The first entry of AB is already not 1:
\((AB)_{11} = 2(-2) + 1(4) + 3(1) = 3 \ne 1\)
(C) BA is defined — Correct. B has 2 columns and A has 2 rows, so BA is defined, of order \(3 \times 3\).
(D) AB and BA both are defined and AB = BA — Incorrect. AB is \(2 \times 2\) and BA is \(3 \times 3\); matrices of different orders cannot be equal.