Linear systems: no, one or infinitely many solutions?
A linear system has either exactly one, no or infinitely many solutions. Which case applies shows up in the row echelon form of the augmented matrix.
How to do it
- 1
Bring the augmented matrix (A|b) to row echelon form.
- 2
A row 0 0 0 | c with c ≠ 0 is a contradiction: no solution, because rank(A) < rank(A|b).
- 3
Otherwise the system is solvable. With as many pivots as unknowns the solution is unique.
- 4
With fewer pivots than unknowns free variables remain: infinitely many solutions. Number of parameters = unknowns − rank.
Example with worked solution
Worked solution
Starting matrix
−1 −4 1 11 −1 −2 0 3 −2 −2 −1 6 1. II − I
−1 −4 1 11 0 2 −1 −8 −2 −2 −1 6 2. III − 2·I
−1 −4 1 11 0 2 −1 −8 0 6 −3 −16 3. III − 3·II
−1 −4 1 11 0 2 −1 −8 0 0 0 8 4. 1/8·III
−1 −4 1 11 0 2 −1 −8 0 0 0 1
rank(A) = 2, rank(A|b) = 3: No solution.
One row reads 0 = c with c ≠ 0, a contradiction.
Your turn
Enter the row operations, e.g. II − 3·I. If you get stuck, "Hint: next step" helps.
How many solutions does the system have?
Bring the augmented matrix to row echelon form and compare rank(A) with rank(A|b).
| 3 | −1 | −2 | −17 |
| 0 | −3 | −1 | −15 |
| −3 | 4 | 5 | 32 |
Your answer
Typical mistakes
- Reading a zero row 0 0 0 | 0 as a contradiction: it always holds and stands for a free parameter.
- Concluding infinitely many solutions from "fewer equations than unknowns": the system can still be inconsistent.
- Forgetting the right-hand side during elimination: a real contradiction then disappears.
Frequently asked questions
What does the Rouché–Capelli theorem say?
Ax = b is solvable exactly when rank(A) = rank(A|b). The solution is unique when this rank equals the number of unknowns.
How do I write down infinitely many solutions?
Make the free variable a parameter, e.g. x₃ = t, and express the other variables in terms of t. The solutions form a line or a plane.
Can a homogeneous system Ax = 0 be unsolvable?
No, x = 0 is always a solution. It has either only this one or infinitely many.