Order the steps of running Gaussian elimination on a 3 by 3 coefficient matrix.

1. Choose a nonzero entry in the first column as the pivot, swapping rows if necessary
2. Add multiples of the pivot row to the rows below to clear the rest of the first column
3. Move to the second column and repeat the process below the first pivot
4. Stop once the matrix is in row echelon form with zero rows at the bottom
5. Count the pivots to read off the rank

Pivot, clear below, move right, repeat, then count. Clearing only below the pivot is what makes the staircase, and the pivot count at the end is the rank, the same number that governs invertibility and the count of free variables.