Gaussian Elimination: Method & Examples

Instructor: Shaun Ault

Shaun is currently an Assistant Professor of Mathematics at Valdosta State University as well as an independent private tutor.

This lesson introduces Gaussian elimination, a method for efficiently solving systems of linear equations using certain operations to reduce a matrix.

What Is Gaussian Elimination?

Gaussian elimination is the process of using valid row operations on a matrix until it is in reduced row echelon form.

I know what you're thinking. Row operations? Reduced row echelon form? We will define these terms as we need them. But first, let's take a look at the reason why we use Gaussian elimination.

Systems of Linear Equations

Suppose you have a system of linear equations.

System of three equations in three variables

How would you go about solving the system? Probably the most efficient way to solve this system (aside from plugging it in a calculator or computer and hitting 'solve') is to follow these three steps:

  1. Translate the system of equations into an augmented matrix.
  2. Use Gaussian elimination (a series of row operations) to reduce the augmented matrix to a simpler form (reduced row echelon form).
  3. Interpret the solution(s) to the system.

This lesson focuses on the second step: the Gaussian elimination. Let's start with row operations.

Row Operations

There are three types of valid row operations that may be performed on a matrix.

  • (OP1) Swap two rows.
  • (OP2) Multiply all entries of a row by a nonzero number.
  • (OP3) Add a multiple of one row to a target row. (Note, the target row is the only row that gets changed in this process.)

It's important to realize that these are just the rules of the game. How we apply them in any given situation will depend on what matrix we are given. Keep in mind: our goal is to transform the matrix into a simpler form, called the reduced row echelon form (RREF), using a series of these row operations.

Reduced Row Echelon Form

We say a matrix is in reduced row echelon form if it satisfies the following requirements:

  • Reading from left to right, the first non-zero entry in any row is 1. This is called the leading entry of the row.
  • The leading entry of a row is always to the right of leading entries in rows above it.
  • Any column with a leading entry has zeros above and below it.

Here is an example matrix in RREF form (not related to our example system above). The leading entries are indicated in bold.

Example RREF matrix

Let's Reduce!

Now that we know the rules of the game (the row operations) and the goal (RREF), it's time to work out an example. Let's assume you know how to find the augmented matrix of the example system mentioned above.

Augmented matrix example

There is already a 1 in the leading entry position for row 1. We need 0s below it. Use OP3 operations. Throughout the following, we use R1 for row 1, R2 for row 2, and R3 for row 3.

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