Question

How do you multiply $3x3$matrix by a $3x1$matrix $?$

Answer 1

To multiply $3x3$matrix by a $3x1$matrix:

Before we multiply two matrices, we have to ensure that the number of columns in the first matrix is equal to the number of rows in another matrix.

Here we have a $3x3$matrix and a $3x1$matrix which is a possible and the resultant matrix is $\mathbf{3}\mathbf{\times }\mathbf{1}$.

Let

$$\begin{gathered} \mathit{A}\mathbf{=}\left[\begin{array}{ccc}1& 2& 3\\ 3& 4& 5\\ 1& 2& 3\end{array}\right]\begin{array}{}\\ \\ \end{array} \\ \mathit{B}\mathbf{=}\left[\begin{array}{c}1\\ 2\\ 3\end{array}\right] \\ \mathit{A}\mathit{B}\mathbf{=}\left[\begin{array}{ccc}\left(1\right).\left(1\right)+& \left(2\right).\left(2\right)+& \left(3\right).\left(3\right)\\ \left(3\right).\left(1\right)+& \left(4\right).\left(2\right)+& \left(5\right).\left(3\right)\\ \left(1\right).\left(1\right)+& \left(2\right).\left(2\right)+& \left(3\right).\left(3\right)\end{array}\right] \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{=}\left[\begin{array}{ccc}1+& 4+& 9\\ 3+& 8+& 15\\ 1+& 4+& 9\end{array}\right] \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{=}\left[\begin{array}{c}14\\ 26\\ 14\end{array}\right] \end{gathered}$$

Hence, multiplication of $3x3$matrix by a $3x1$matrix is possible as stated above.