The enlargement principle provides techniques for inverting any nonsingular matrix by building the inverse upon the inverses of successively larger submatrices. The computing routines are relatively easily learned since they are repetitive. Three different enlargement routines are outlined: first-order, second-order, and geometric. None of the procedures requires much more work than is involved in squaring the matrix.
"Enlargement Methods for Computing the Inverse Matrix." Ann. Math. Statist. 17 (3) 336 - 343, September, 1946. https://doi.org/10.1214/aoms/1177730946