# Matrix multiplication: is $\begin{bmatrix} 3 \\ 2 \\ 1 \\ \end{bmatrix}\begin{bmatrix} 1 & 2 & 3 \\ \end{bmatrix}$ defined?

Is this matrix multiplication is possible? Microsoft mathematics gives me an answer for it? How it could be a correct? If we need multiple two matrix, number of rows and columns should be equal. So how it is possible?

$$\begin{bmatrix} 3 \\ 2 \\ 1 \\ \end{bmatrix}\begin{bmatrix} 1 & 2 & 3 \\ \end{bmatrix}$$

## migrated from mathematica.stackexchange.comSep 1 '14 at 7:05

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Yes, the number of columns in the left matrix equals the number of rows in the right matrix, so it's defined. We have: $$\begin{bmatrix} 3 \\ 2 \\ 1 \\ \end{bmatrix}_{3 \times \color{blue}{\mathbf 1}} \begin{bmatrix} 1 & 2 & 3 \\ \end{bmatrix}_{\color{blue}{\mathbf 1} \times 3} =\begin{bmatrix} 3 & 6 & 9 \\ 2 & 4 & 6 \\ 1 & 2 & 3 \\ \end{bmatrix}.$$
By definition, in multiplying $[A_{ij}]_{k \times \color{blue}{\mathbf n}}$ with $[B_{ij}]_{\color{blue}{\mathbf n} \times m}$, the entry in cell $(r,c)$ is $$A_{r1}B_{1c}+A_{r2}B_{2c}+\cdots+A_{rn}B_{nc}.$$ In the above case, we have $n=1$.