nmod_mat_reduce_row

P has one entry per column: P[i] is the index of the row whose pivot (leading entry, which must be 1) is in column $i$, or $-1$ if there is none. Row $m$ is reduced from left to right by subtracting multiples of these rows. At the first column $i$ where it has a nonzero entry and P[i] is $-1$, the row is divided by that entry, P[i] is set to $m$ and $i$ is returned. If the row becomes zero, $-1$ is returned. L[j] is the number of leading columns of row $j$ that may be nonzero; entries beyond it are taken to be zero, which saves work when the rows get longer towards the bottom. Setting every entry of L to the number of columns is always valid. The pivot rows may be any rows of $M$, but at most $m$ of them; the updates are accumulated without reduction in up to three words. This is the elimination step of nmod_mat_minpoly.