Gaussian Elimination Summary

Row echelon form (REF)
For each non-zero row, the leading entry is to the right of the leading entry of the row above.

E.g. $\begin{pmatrix} 0 & \mathbf{1} & 7 & 2\\ 0 & 0 & \mathbf{9} & 3\\ 0 & 0 & 0 & 0 \end{pmatrix}$

Note that the leading entry 9 of the second row is to the right of the leading entry 1 of the first row.

Reduced row echelon form (RREF)
A row echelon form is said to be reduced, if in each of its pivot columns, the leading entry is 1 and all other entries are 0.

E.g. $\begin{pmatrix} 1 & 0 & 0 & 2\\ 0 & 1 & 0 & 3\\ 0 & 0 & 1 & 4 \end{pmatrix}$

Elementary Row Operations
1) $cR_i$ — multiply the $i$th row by the constant $c$
2) $R_i \leftrightarrow R_j$ — swap the $i$th and the $j$th row
3) $R_i+cR_j$ — add $c$ times of the $j$th row to the $i$th row.

Gaussian Elimination Summary
Gaussian Elimination is essentially using the elementary row operations (in any order) to make the matrix to row echelon form.

Gauss-Jordan Elimination
After reaching row echelon form, continue to use elementary row operations to make the matrix to reduced row echelon form.