If matrices \(A =\begin{bmatrix}1&2&3\end{bmatrix}\) and \(B =\begin{bmatrix}1\\2\\3\end{bmatrix}\), then BA is equal to
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(\begin{bmatrix}1&2&3\\2&4&6\\3&6&9\end{bmatrix}\)
B is of order 3 × 1 and A is of order 1 × 3, so BA is a 3 × 3 matrix. Its (i, j) entry is the i-th entry of B times the j-th entry of A.
\(\displaystyle BA = \begin{bmatrix}1\\2\\3\end{bmatrix}\begin{bmatrix}1&2&3\end{bmatrix}\)
\(\displaystyle = \begin{bmatrix}1\times1&1\times2&1\times3\\2\times1&2\times2&2\times3\\3\times1&3\times2&3\times3\end{bmatrix}\)
\(\displaystyle = \begin{bmatrix}1&2&3\\2&4&6\\3&6&9\end{bmatrix}\)
Why the other options are incorrect
Option (1) \(\begin{bmatrix}14\end{bmatrix}\) — Incorrect. This is AB = [1 × 1 + 2 × 2 + 3 × 3] = [14], a 1 × 1 matrix. Matrix multiplication is not commutative, and BA is 3 × 3.
Option (2) \(\begin{bmatrix}1&4&9\end{bmatrix}\) — Incorrect. This lists the products of matching entries (1 × 1, 2 × 2, 3 × 3), which is not a matrix product; BA is 3 × 3, not 1 × 3.
Option (4) \(\begin{bmatrix}1\\4\\9\end{bmatrix}\) — Incorrect. The same products of matching entries, written as a column; BA is 3 × 3, not 3 × 1.