Matrix multiplication
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} We can multiply this matrix by \lt math\gt 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 \dot 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} 0.62 \cdot 0 & 0.62\cdot 340 & 0.62 \dot 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} }[/math]