Percolated+News

Example—Gauss Jordan Method

Solve the following system of equations:

Your goal is to obtain the following:

 

**Use the multiplicative inverse to obtain 1 and the additive inverse to obtain 0.

First, create an augmented matrix as follows:

Step 1:  We need 1 in the position Row 1, Column 1.  To obtain this, we multiply the original Row 1 by   

The new first row looks like this:

 

Step 2:  We need 0 in the position Row 2, Column 1.  To obtain this, we multiply the new Row 1 by 2 and add to the original Row 2.  The new Row 2 looks like this:

Step 3:  We need 0 in the position Row 3, Column 1.  To obtain this, we multiply the new Row 1 by -3 and add to the original Row 3.  The new Row 3 looks like this:

**Our matrix now looks like the following:

Step 4:  We need a 1 in position Row 2, Column 2.  To obtain this, we multiply the previous Row 2 by   The new Row 2 looks like this:

Step 5:  We need a 0 in position Row 1, Column 2.  To obtain this, we multiply the new Row 2 by  and add to the previous Row 1.  The new Row 1 looks like this:

Step 6:  We need a 0 in position Row 3, Column 2.  To obtain this, we multiply the new Row 2 by  and add to the previous Row 3.  The new Row 3 looks like this:

*The matrix should now look like the following:

Step 7:  We now need a 1 in position Row 3, Column 3.  To obtain this, we multiply the previous Row 3 by -4.  The new Row 3 looks like the following:

Step 8:  We now need a 0 in position Row 2, Column 3.  To obtain this, we multiply the new Row 3 by  and add to the previous Row 2.  The new 2^nd row looks like the following:

Step 9:  We now need a 0 in position Row 1, Column 3.  To obtain this, we multiply the new Row 3 by  and add to the previous Row 1.  The new Row 1 looks like the following:

**Finally, our matrix looks like the following:

This implies that x = 1, y = 2, and z = 1.  You can check by substituting these values into the original system of equations.