Augmented matrices and systems of linear equations

You can think of an augmented matrix as being a way to organize the important parts of a system of linear equations. These “important parts” would be the coefficients (numbers in front of the variables) and the constants (numbers not associated with variables).


Writing the augmented matrix for a system

Let’s look at two examples and write out the augmented matrix for each, so we can better understand the process. The key is to keep it so each column represents a single variable and each row represents a single equation. The augment (the part after the line) represents the constants.


Write the augmented matrix for the system of equations:
\(\begin{array}{l}3x_1 + 5x_2 – x_3 = 10\\ x_1 + 4x_2 + x_3 = 7\\ 9x_1 + 2x_3 = 1\\ \end{array}\)


There are three variables, and so we will need a column for each. Be careful – notice that the last equation doesn’t have an \(x_2\). That will be represented with a 0.

Augmented matrix for the system; each column represents the coefficients for the variables and each row represents an equation. Row 1 is 3 5 -1 10, Row 2 is 1 4 1 7, and row 3 is 9 0 2 1. The 10, 7, and 1 each represent the constants from the equations.


Write the augmented matrix for the system of equations:

\(\begin{array}{l} x_1 – 2x_2 + 8x_3 + x_4 + x_5 = 2\\ 3x_1 – x_2 + x_3 + 2x_4 + 2x_5 = -3\\ \end{array} \)


Even though this is not the type of system we are used to seeing in our usual algebra classes, we can still write an augmented matrix to represent it. The augmented matrix for this system would be:

Augmented matrix for example 2. The first row is 1 -2 8 1 1 2, for the coefficients and constants from the first equation. The second row is 3 -1 1 2 2 -3 for the constants and coefficients from the second equation.

Common Questions

Does the order that I write the rows in matter?

No. In algebra, when you were solving a system like \(3x + y = 5\) and \(2x + 4y = 7\), it didn’t matter if you wrote one equation first or second. The solution to the problem didn’t change. The same is true when you have more than two equations. Since each row represents an equation, the order that you write the rows in doesn’t matter.

What is this used for?

Putting a system of equations in this form will allow us to use a new idea called row operations to find its solution (if one exists), describe the solution set (when there are infinitely many solutions), and more. Row operations can help us organize a way to do this regardless of how many variables or how many equations we are given. This will be studied in later articles.