Let \(A = \begin{bmatrix}1\\-2\\3\end{bmatrix}\) and \(B = \begin{bmatrix}2&0&-1\end{bmatrix}\).
(A) AB is a matrix of order \(3 \times 3\).
(B) \(BA = \begin{bmatrix}-1\end{bmatrix}\)
(C) AB is a symmetric matrix.
(D) The element in the second row, third column of AB is 2.
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → (A), (B) and (D) only
A is of order \(3 \times 1\) and B is of order \(1 \times 3\). Each element of AB is one element of A times one element of B:
\( \displaystyle AB = \begin{bmatrix}2&0&-1\\-4&0&2\\6&0&-3\end{bmatrix} \)
BA is of order \(1 \times 1\):
\(BA = \begin{bmatrix}2(1) + 0(-2) + (-1)(3)\end{bmatrix} = \begin{bmatrix}-1\end{bmatrix}\)
So AB is the \(3 \times 3\) matrix above, with first-row, second-column element 0 and second-row, first-column element \(-4\), and \(BA = \begin{bmatrix}-1\end{bmatrix}\).
Checking the statements
(A) AB is a matrix of order \(3 \times 3\). — Correct. \((3 \times 1)(1 \times 3) \rightarrow 3 \times 3\).
(B) \(BA = \begin{bmatrix}-1\end{bmatrix}\) — Correct. \(2 + 0 - 3 = -1\).
(C) AB is a symmetric matrix. — Incorrect. The elements in positions (1, 2) and (2, 1) are 0 and \(-4\).
(D) The element in the second row, third column of AB is 2. — Correct. \((-2)(-1) = 2\).