Case of a nonsquare matrix

If the matrix is not square, then we may still write where is still a permutation matrix and is still a (special) lower triangular matrix. However, the matrix needs a new description. There are still pivots in for each nonzero row and the pivotal columns successively march to the right as the pivots move down. However, there could be some extra columns interspersed between the pivotal columns. The only requirement is that a nonpivotal column cannot have nonzero entries lower than any earlier pivotal column .
The only way to understand this legalistic statement is to study some examples.
Given below are some valid examples of such matrices. The form is called a row echelon form .
The pivots are marked in yellow and nonpivotal columns are in cyan color. They represent coefficients of the free variables.

> showmat(matrix(3,4,[1,2,3,4,0,1,1,2,0,0,1,3]),[[piv,[1,1]],[piv,[2,2]],[piv,[3,3]],[col,4,cyan]]):

> showmat(matrix(3,4,[1,2,3,4,0,1,1,2,0,0,0,0]),[[piv,[1,1]],[piv,[2,2]],[col,3,cyan],[col,4,cyan]]):

> showmat(matrix(3,4,[1,2,3,4,0,0,1,2,0,0,0,0]),
[[piv,[1,1]],[piv,[2,3]],[col,2,cyan],[col,4,cyan]]):

> showmat(matrix(3,2,[0,1,0,0,0,0]),[[piv,[1,2]],[col,1,cyan]]):

>