Gaussian elimination
The goal of Gaussian elimination is to get the matrix in row-echelon form. If a matrix is in row-echelon form, that means that reading from left to right, each row will start with at least one more zero term than the row above it. Some definitions of Gaussian elimination say that the matrix result has to be in reduced row-echelon form. That means that the matrix is in row-echelon form and the only non-zero term in each row is 1. Gaussian elimination that creates a reduced row-echelon matrix result is sometimes called Gauss-Jordan elimination.