[Gilbert Strang] Gauss-Jordan Elimination

For example, let's solve the following system of linear equations.

We can rewrite this system in matrix form as follows.

This can be expressed more compactly using the augment matrix.

This is called an augmented matrix.
When performing Gaussian elimination, we use the following elementary row operations.

1. Multiply a row by a non-zero scalar.
2. Swap two rows.
3. Add a multiple of one row to another row.

Implementation of Gaussian Elimination

Step 1: Start with the augmented matrix corresponding to the system.

Step 2: Multiply the first row by -5 to make the pivot for elimination easier when adding to the second row.

Step 3: Add the second row to the modified first row (Row 1 + Row 2).
This gives a new first row of [0, 12, 1], while the second row remains unchanged.
The resulting matrix is

Now, we can get solution in linear equation.