If \( A=\begin{bmatrix}1 & 2 & -3\end{bmatrix} \) and \( B=\begin{bmatrix}1 & 3\\7 & -8\\1 & 2\end{bmatrix} \), then which of the following are true?
(A) AB is a \( 2\times 3 \) matrix
(B) \( \mathrm{B}^2 \) is a \( 3\times 3 \) matrix
(C) BA is not possible
(D) \( \mathrm{A}^3 \) is not possible
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (C) and (D) only
A is of order \( 1\times 3 \) and B is of order \( 3\times 2 \). A product of an \( m\times n \) matrix by a \( p\times q \) matrix exists only when n = p, and then its order is \( m\times q \).
\( \mathrm{AB}:\ (1\times 3)(3\times 2)=1\times 2 \)
\( \mathrm{BB}:\ (3\times 2)(3\times 2) \), not defined (2 ≠ 3)
\( \mathrm{BA}:\ (3\times 2)(1\times 3) \), not defined (2 ≠ 1)
Powers exist only for square matrices, and A is not square.
So AB is \( 1\times 2 \), while \( \mathrm{B}^2 \), BA and \( \mathrm{A}^3 \) do not exist.
Checking the statements
(A) AB is a \( 2\times 3 \) matrix — Incorrect. AB is of order \( 1\times 2 \).
(B) \( \mathrm{B}^2 \) is a \( 3\times 3 \) matrix — Incorrect. B has 2 columns but 3 rows, so \( \mathrm{B}^2 \) does not exist.
(C) BA is not possible — Correct. B has 2 columns and A has 1 row.
(D) \( \mathrm{A}^3 \) is not possible — Correct. A is \( 1\times 3 \), not square.