Augmented Matrices
This page follows the JoeCMath video Convert a System of Equations into an Augmented Matrix. The video starts with a system already written in a clean order, then works through a second system where the equations need to be rearranged before the augmented matrix is built.
Main Idea
An augmented matrix stores the coefficients from a system of linear equations, with the constants placed in the last column.
For a system in the variables $x$, $y$, and $z$,
\[\begin{aligned} a_1x+b_1y+c_1z&=d_1 \\ a_2x+b_2y+c_2z&=d_2 \\ a_3x+b_3y+c_3z&=d_3, \end{aligned}\]the augmented matrix is
\[\left[ \begin{array}{ccc|c} a_1 & b_1 & c_1 & d_1 \\ a_2 & b_2 & c_2 & d_2 \\ a_3 & b_3 & c_3 & d_3 \end{array} \right].\]Each equation becomes one row. Each variable position becomes one coefficient column. The constants go after the vertical bar.
Watch this section: introduction at 0:00.
First System
The first system in the video is already arranged in the same variable order:
\[\begin{aligned} 2x-3y-5z&=10 \\ 5x+2y+7z&=12 \\ 6x-2y+10z&=0. \end{aligned}\]Since the variables appear in the order $x$, $y$, $z$ in every equation, the coefficient columns can be read straight across:
\[\left[ \begin{array}{ccc|c} 2 & -3 & -5 & 10 \\ 5 & 2 & 7 & 12 \\ 6 & -2 & 10 & 0 \end{array} \right].\]The first row comes from $2x-3y-5z=10$. The second row comes from $5x+2y+7z=12$. The third row comes from $6x-2y+10z=0$.
Watch this section: first augmented matrix setup at 0:08.
What the Columns Mean
The video keeps the columns tied to the variables:
| Column | Meaning |
|---|---|
| First coefficient column | Coefficients of $x$ |
| Second coefficient column | Coefficients of $y$ |
| Third coefficient column | Coefficients of $z$ |
| Last column | Constants on the right side |
Watch this section: coefficients and constants at 0:42.
Disordered System
The second system in the video is not as ready to copy:
\[\begin{aligned} 6x-3y-5z&=5 \\ 1&=-y+9z \\ 8y+3x&=-1. \end{aligned}\]Before building the augmented matrix, the video rewrites the equations so the variables line up in the same order in each row.
The first equation is already in $x$, $y$, $z$ order:
\[6x-3y-5z=5.\]The second equation has no $x$ term, so place a $0x$ in front:
\[0x-y+9z=1.\]The third equation is written as $8y+3x=-1$, so rewrite it in $x$, $y$, $z$ order and include the missing $z$ term:
\[3x+8y+0z=-1.\]Watch this section: disordered system starts at 1:54.
Second Augmented Matrix
After the equations are rewritten, the system is
\[\begin{aligned} 6x-3y-5z&=5 \\ 0x-y+9z&=1 \\ 3x+8y+0z&=-1. \end{aligned}\]Now each row has the same variable positions, so the augmented matrix is
\[\left[ \begin{array}{ccc|c} 6 & -3 & -5 & 5 \\ 0 & -1 & 9 & 1 \\ 3 & 8 & 0 & -1 \end{array} \right].\]The zeros hold the place for missing variables so the columns still mean $x$, $y$, and $z$ all the way down.
Watch this section: insert zeros for missing variables at 3:00.
Timestamp Guide
Related Linear Algebra Work
Augmented matrices are often the setup step before row reduction. For what comes next, use the Elementary Row Operations companion or the Row Echelon Form companion.
You can also browse the Matrices topic page or use the JoeCMath Matrix/Linear Algebra playlist.