Matrix multiplication
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We start with a simple example showing a practical use of multiplying a matrix by numbers (scalars). Below is an example of a distance matrix between three cities, in kilometers:
[math]\displaystyle{ A =\begin{bmatrix} 0 & 340 & 360 \\ 340 & 0 & 450 \\ 360 & 450 & 360 \end{bmatrix} }[/math]
We can multiply this matrix by [math]\displaystyle{ 0.62 }[/math] to (approximately) convert it to distances in miles:
[math]\displaystyle{ B =0.62\cdot A \begin{bmatrix} 0.62 \cdot 0 & 0.62\cdot 340 & 0.62 \cdot 360 \\ 0.62 \cdot 340 & 0.62\cdot 0 & 0.62\cdot 450 \\ 0.62 \cdot 360 & 0.62\cdot 450 & 0.62\cdot 360 \end{bmatrix} = \begin{bmatrix} 0 & 210.8 & 223.2 \\ 210.8 & 0 & 279 \\ 223.2 & 279 & 0 \end{bmatrix} }[/math]