Three possible outcomes
After elimination, a linear system usually falls into one of three types:
- one solution,
- infinitely many solutions,
- no solution.
The reduced matrix tells you which case you are in.
Definition
Solution-set types
The reduced form of a linear system shows whether the system has one solution, infinitely many solutions, or no solution.
How to read the pattern
- If every variable has a pivot, the system has a single solution.
- If at least one variable is free and there is no contradiction row, the system has infinitely many solutions.
- If a contradiction row appears, the system has no solution.
Worked example
Recognize the three cases
Consider these reduced matrices:
has one solution.
has infinitely many solutions because one variable is free.
has no solution because the last row is a contradiction.
Pause here and compare the three reduced matrices below. Try to name the case only after you have looked for pivots, free variables, and contradiction rows.
Try it here
Solution-set classifier
The live classifier compares three representative reduced matrices and explains what each structure means.
| 1 | 0 | 0 | 2 |
| 0 | 1 | 0 | -1 |
| 0 | 0 | 1 | 3 |
Why it works
Every variable is a pivot variable, so the system has one solution.
Common mistake
Common mistake
Free variable does not mean unfinished work
A free variable means the solution set has a parameter. It does not mean the problem is broken or incomplete.
Quick check
Quick check
What does `[0 0 0 | 1]` tell you about the system?
Solution
Answer
Prerequisite link
This page uses the ideas from 2.3 Gaussian elimination and RREF.