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 ith row by the constant c
2) R_i \leftrightarrow R_j — swap the ith and the jth row
3) R_i+cR_j — add c times of the jth row to the ith 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.