Gauss-Jordan Elimination

Just like the Gaussian Elimination, Gauss- jordan elimination aims to solve a system of linear equations. This is also done by transforming the system into augmented matrix, but the difference is, it needs to be a reduced row-echelon form.

Reduced Row-Echelon Form
- is basically a simpler form of a matrix. A matrix is in reduced row-echelon form if:

1. If there is a row where every entry is zero, then this row lies below any other row that contains a nonzero entry.
2. The leftmost nonzero entry of a row is equal to 1.
3. The leftmost nonzero entry of a row is the only nonzero entry in its column.
4. The nonzero entry on the higher row should be further to the left than the nonzero entry on the lower row.

     See that the final augmented matrix, the nonzero entries are in a staircase pattern. That means the equivalent linear equations are x=1, y=4 and z=-2 because it's the only nonzero entries in that row.

    

Gaussian Elimination

Gaussian Elimination is performed to certain augmented matrices so that they can be in a row-echelon form. 

Steps
*From a System, transform it to an augmented matrix. 
*Perform Elementary Row Operations to transform it to Row-Echelon form.
*Do back-substitution.

3 Elementary Row Operations
1. Interchanging 2 rows.

2. Multiplying a row by a nonzero constant.
3. Adding a multiple of a row to another row. 

How to Know if the matrix is in Row-Echelon Form or not?

a. Rows with zero entries must be at the bottom of the matrix. 

b. All leading coefficients must be 1.
c. The leading coefficient of the higher row must be further to the left than the leading coefficient of the lower row.