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It's useful to have knowledge of them. They tend to pop up in random places. For instance, even simple recurrence relations like the Fibonacci one can be computed in logarithmic time via matrix exponentiation.

Linear algebra in general is completely critical in machine learning as well.

I use them regularly in all sorts of unexpected places. The most common is probably machine learning, but I also use them in safety critical computations of projected courses, ETAs of various processes, and computing optimal geometries.

Any time you're dealing with linear transformations of vectors (ie functions f such that f(u + v) = f(u) + f(v) and f(a * u) = a * f(u) for any vectors u,v and scalar a), then you're implicitly dealing with matrices.

The really huge Google Matrix from the world of Information Retrieval: http://en.wikipedia.org/wiki/Google_matrix