Inverting a lower triangular matrix

Here's a 120x120 integer-valued lower triangular matrix with 1s on the diagonal:

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Trying to invert it takes more than 5Gb (didn't try beyond that, killed the process after a few minutes).
Trying to invert it after transposing it works instantly.
I think we can do a little better in our matrix inversion algorithm...

The issue provides a 120x120 lower triangular matrix example but names no source files or tests. Start by reproducing the matrix inversion in Macaulay2, then identify the relevant inversion entry point and define the expected result before adding a focused regression test.