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Question

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

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Solution

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 $\mathbf{3}\mathbf{×}\mathbf{1}$.Let $\mathbit{A}\mathbf{=}\left[\begin{array}{ccc}1& 2& 3\\ 3& 4& 5\\ 1& 2& 3\end{array}\right]\begin{array}{}\\ \\ \end{array}\phantom{\rule{0ex}{0ex}}\mathbit{B}\mathbf{=}\left[\begin{array}{c}1\\ 2\\ 3\end{array}\right]\phantom{\rule{0ex}{0ex}}\mathbit{A}\mathbit{B}\mathbf{=}\left[\begin{array}{ccc}\left(1\right).\left(1\right)+& \left(2\right).\left(2\right)+& \left(3\right).\left(3\right)\\ \left(3\right).\left(1\right)+& \left(4\right).\left(2\right)+& \left(5\right).\left(3\right)\\ \left(1\right).\left(1\right)+& \left(2\right).\left(2\right)+& \left(3\right).\left(3\right)\end{array}\right]\phantom{\rule{0ex}{0ex}}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\left[\begin{array}{ccc}1+& 4+& 9\\ 3+& 8+& 15\\ 1+& 4+& 9\end{array}\right]\phantom{\rule{0ex}{0ex}}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\left[\begin{array}{c}14\\ 26\\ 14\end{array}\right]\phantom{\rule{0ex}{0ex}}$Hence, multiplication of $3x3$matrix by a $3x1$matrix is possible as stated above.

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