An arbitrary matrix can be solved using Gauss-Jordan elimination with O(n^3) complexity. A tridiagonal matrix (i.e. a special type of matrix) can be solved using Gauss-Jordan with O(n) complexity. That is, a for an arbitrary matrix, Gauss-Jordan is a cumbersome algorithm, but for a tridiagonal matrix, the algorithm can be expressed as a loop and a simple formula.

If I were writing, for example, a compiler or some optimization program, is there a way to test the problem to see if it has become less complex, so I can express the algorithm more simply?

The most straight-forward approach would probably be to traverse all the matrix elements that belong to each "triangle" to determine if they are all zero. For large matrices I imagine that this operation would definitely pay off if they do in fact turn out to be tridiagonal, however since I have never done this myself I'm afraid this assumption is just conjecture.
But I would like to hear about the results if you do program this

The (kind of) problems you've described has known beforehand loop limits, so the loops can be unrolled statically. Counting lines in the unrolled loop is a pretty good measure of complexity, though, strictly saying, you should also count arithmetic operations.