Multiplying matrices is useful in lots of engineering applications, but the one that comes to my mind is in computer graphics. Which happens to be the answers to our system of linear equations! Which simplifies (thanks to our "weird" definition of multiplication) to : To balance the equation out (do the same stuff to both sides), you multiply both sides by the inverse of the first matrix to cancel it out on the left side (i've taken an extremely simple-to-inverse example on purpose, inversing is pretty complicated for many other matrices, often indicating that your system of equations is not going to have nice integers as answers): Matrix multiplication can be used to solve linear equations: It is hard to explain without going into linear algebra and vector spaces, but maybe this image helps to give some intuition: Matrices represent linear functions between vector spaces.
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