Gaussian Elimination

Gaussian Elimination

We've already seen how to solve a linear system using inverse matrices. We now introduce a new method for solving linear systems called Gaussian Elimination.

The plan is to put the linear system into an augmented matrix, that is, a matrix which contains not only the coefficients but the constants as well. We then use row operations to reduce the linear system to reduced row echelon form.

Here is what reduced row echelon form looks like

The left-hand, square, part of the matrix is identical to the 2X2 identity matrix while the far right column has two constants in it.

Consider the linear system

This goes into an augmented matrix as

Notice that the left-hand square part of the matrix contains the coefficients of the variables while the right column contains the constants.

We now want to use row operations to change the left-hand side into the 2x2 identity matrix. The standard procedure is to work from the far left column and move right. First get the 1 in the correct position, then use it to eliminate the numbers above and below it, i.e. change them into 0s.

Let's divide the first row by 2.

Now we add -3 times the first row to the second row thereby putting a 0 in the 2,1 position.

Now move to the second column. Multiply the second row by -2/7

Now, add -1/2 times the second row to the first

This matrix is in reduced row echelon form. We can now read off the equations

and

You may think this is unnecessarily complicated. However, it does have the added benefit of working quite well for larger systems. It is also useful when the system has more than one solution.

Example: Solve the linear system

This goes into the augmented matrix

Here's a possible sequence of steps:

We can now see we have two identical rows. This tells us that the three equations were really two equations in disguise. We can easily eliminate the third row by