Note that two matrices (of the same dimensions) can be transformed into each other if and only if they have the same rank.

A (non-optimal, but straightforward) procedure for doing so is like so: First, use Gaussian elimination row-wise to put any matrix into reduced row echelon form. One can now use Gaussian elimination column-wise to transform the matrix into a 2x2 block matrix whose upper-left block is an identity matrix (of size corresponding to the rank) and whose other blocks are zero. Since all moves are invertible, any two matrices of the same rank are thus connected via the same such block matrix.

In general, it is necessary to use both row and column moves. However, if both matrices are square with full rank (as in today's puzzle), one can just use row moves (or just as well, just use column moves), using just Gaussian elimination. More generally, one can just use row moves iff both matrices have the same row space, and similarly for columns.

This implicitly relies on row and column rank being the same.