Mechanical engineers use matrices a lot. If you want to see how heat will spread through a material like your computer processor/heat sink then you use a matrix to represent the temperature at a bunch of points all over the surface. If you do it right then this allows your computer to calculate the temperature at each and every point in one go. It is more than just storing data, it allows you to do special math.
A very similar method is used to simulate materials and how they bend like if you were designing a bracket or even a whole bridge.
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