Wrote up an LU factorization algorithm for a square matrix of arbitrary size, and observed it to work pretty well on several small-sized matrices (n < 10). I expect this algorithm to perform reasonably on larger-sized matrices, though I still need to run some confirmation testing using larger-sized matrices (it does O(n^2) comparisons and 2n^3/3 flops). Once that’s done, I can then use this LU factorization algorithm as the building block for developing a matrix inversion algorithm.
I was somewhat frustrated by how the majority of matrix-related resources I’ve checked avoid showing specifics of matrix inversion algorithm. The reasoning is that such algorithm would be highly inefficient and one should consider directly solving a system of linear equations instead of inverting a matrix first then solving it. I have no doubt what they say is true, but the sad fact is that, sometimes one may need to explicitly compute an inverse matrix as part of another larger algorithm. :(