From equations to rows
When we write a linear system as a matrix, each equation becomes one row. The coefficients go first, and the constants go in the last column.
That full table is called an augmented matrix.
Definition
Augmented matrix
An augmented matrix places the coefficient matrix and the constant column in one table so you can row-reduce the whole system at once.
The three row operations
The local notes use three elementary row operations:
- swap two rows,
- multiply a row by a nonzero number,
- add a multiple of one row to another row.
These operations preserve the solution set, so they are safe for solving a system.
Worked example
Turn a system into an augmented matrix
Consider
Its augmented matrix is
Now use the row operation R_2 \leftarrow R_2 - 3R_1 to begin eliminating the
x-term in the second row.
Why the row operations are allowed
Row operations are just a matrix version of equivalent equation operations. They do not change which number lists satisfy the system.
Use the explorer below while the three operations are still fresh. The point is to see that editing a row means editing one equation in a controlled way, not scrambling the problem.
Try it here
System-to-augmented-matrix explorer
The live explorer highlights how each equation becomes one matrix row plus one constant entry.
System
- x + 2y = 5
- 3x - y = 4
Result
| 1 | 2 | 5 |
| 3 | -1 | 4 |
Common mistake
Common mistake
Do not change only the constants
If you change one side of an equation but not the other, you are no longer working with an equivalent system.
Quick check
Quick check
Which row operation keeps the solution set unchanged?
Solution
Answer
Prerequisite link
This page builds on 1.1 Equations and solution sets and 2.1 Matrix basics.