The Augmented Matrix of a System of Equations

A matrix can serve as a device for representing and solving a system of equations. To express a system in matrix form, we extract the coefficients of the variables and the constants, and these become the entries of the matrix. We use a vertical line to separate the coefficient entries from the constants, essentially replacing the equal signs. When a system is written in this form, we call it an augmented matrix.

For example, consider the following [latex]2\times 2[/latex] system of equations.

[latex]\begin{array}{l}3x+4y=7\\ 4x - 2y=5\end{array}[/latex]

We can write this system as an augmented matrix:

[latex]\left[\begin{array}{rr}\hfill 3& \hfill 4\\ \hfill 4& \hfill -2\end{array}\text{ }|\text{ }\begin{array}{r}\hfill 7\\ \hfill 5\end{array}\right][/latex]

We can also write a matrix containing just the coefficients. This is called the coefficient matrix.

[latex]\left[\begin{array}{cc}3& 4\\ 4& -2\end{array}\right][/latex]

A three-by-three system of equations such as

[latex]\begin{array}{l}3x-y-z=0\hfill \\ \text{ }x+y=5\hfill \\ \text{ }2x - 3z=2\hfill \end{array}[/latex]

has a coefficient matrix

[latex]\left[\begin{array}{rrr}\hfill 3& \hfill -1& \hfill -1\\ \hfill 1& \hfill 1& \hfill 0\\ \hfill 2& \hfill 0& \hfill -3\end{array}\right][/latex]

and is represented by the augmented matrix

[latex]\left[\begin{array}{rrr}\hfill 3& \hfill -1& \hfill -1\\ \hfill 1& \hfill 1& \hfill 0\\ \hfill 2& \hfill 0& \hfill -3\end{array}\text{ }|\text{ }\begin{array}{r}\hfill 0\\ \hfill 5\\ \hfill 2\end{array}\right][/latex]

Notice that the matrix is written so that the variables line up in their own columns: x-terms go in the first column, y-terms in the second column, and z-terms in the third column. It is very important that each equation is written in standard form [latex]ax+by+cz=d[/latex] so that the variables line up. When there is a missing variable term in an equation, the coefficient is 0.

How To: Given a system of equations, write an augmented matrix.

  1. Write the coefficients of the x-terms as the numbers down the first column.
  2. Write the coefficients of the y-terms as the numbers down the second column.
  3. If there are z-terms, write the coefficients as the numbers down the third column.
  4. Draw a vertical line and write the constants to the right of the line.

Example 1: Writing the Augmented Matrix for a System of Equations

Write the augmented matrix for the given system of equations.

[latex]\begin{array}{l}\text{ }x+2y-z=3\hfill \\ \text{ }2x-y+2z=6\hfill \\ \text{ }x - 3y+3z=4\hfill \end{array}[/latex]

Solution

The augmented matrix displays the coefficients of the variables, and an additional column for the constants.

[latex]\left[\begin{array}{rrr}\hfill 1& \hfill 2& \hfill -1\\ \hfill 2& \hfill -1& \hfill 2\\ \hfill 1& \hfill -3& \hfill 3\end{array}\text{ }|\text{ }\begin{array}{r}\hfill 3\\ \hfill 6\\ \hfill 4\end{array}\right][/latex]

Try It 1

Write the augmented matrix of the given system of equations.

[latex]\begin{array}{l}4x - 3y=11\\ 3x+2y=4\end{array}[/latex]

Writing a System of Equations from an Augmented Matrix

We can use augmented matrices to help us solve systems of equations because they simplify operations when the systems are not encumbered by the variables. However, it is important to understand how to move back and forth between formats in order to make finding solutions smoother and more intuitive. Here, we will use the information in an augmented matrix to write the system of equations in standard form.

Example 2: Writing a System of Equations from an Augmented Matrix Form

Find the system of equations from the augmented matrix.

[latex]\left[\begin{array}{rrr}\hfill 1& \hfill -3& \hfill -5\\ \hfill 2& \hfill -5& \hfill -4\\ \hfill -3& \hfill 5& \hfill 4\end{array}\text{ }|\text{ }\begin{array}{r}\hfill -2\\ \hfill 5\\ \hfill 6\end{array}\right][/latex]

Solution

When the columns represent the variables [latex]x[/latex], [latex]y[/latex], and [latex]z[/latex],

[latex]\left[\begin{array}{rrr}\hfill 1& \hfill -3& \hfill -5\\ \hfill 2& \hfill -5& \hfill -4\\ \hfill -3& \hfill 5& \hfill 4\end{array}\text{ }|\text{ }\begin{array}{r}\hfill -2\\ \hfill 5\\ \hfill 6\end{array}\right]\to \begin{array}{l}x - 3y - 5z=-2\hfill \\ 2x - 5y - 4z=5\hfill \\ -3x+5y+4z=6\hfill \end{array}[/latex]

Try It 2

Write the system of equations from the augmented matrix.

[latex]\left[\begin{array}{ccc}1& -1& 1\\ 2& -1& 3\\ 0& 1& 1\end{array}|\begin{array}{c}5\\ 1\\ -9\end{array}\right][/latex]