Linear Algebra
Chapter 1/Matrices and Systems of Equations

1.2Row Echelon Form

Elimination doesn't always reach a nice triangle. Row echelon form is what you get in general, and reading it correctly tells you instantly whether the system is inconsistent, uniquely solvable, or has a family of solutions.

What you must be able to do

  • Row echelon form: leading entry of each nonzero row is , leading zeros strictly increase going down, zero rows at the bottom.
  • A row ⟹ inconsistent. Otherwise consistent.
  • Columns with pivots ⟹ lead variables; columns without ⟹ free variables. Number of free variables .
  • Homogeneous systems are always consistent; if they have nontrivial solutions (Thm 1.2.1).
Definition · Row echelon form
A matrix is in row echelon form if (i) the first nonzero entry of each nonzero row is ; (ii) the number of leading zeros strictly increases from each row to the next; (iii) all-zero rows sit at the bottom. Gaussian elimination is the process of getting there.
Definition · Reduced row echelon form
Additionally, each leading is the only nonzero entry in its column. Getting there is Gauss–Jordan reduction. This is the form you want when there are free variables, because the solution can be read off directly.

Variables whose columns contain a leading are lead variables; the rest are free variables. Free variables get assigned arbitrary parameters, and the lead variables are then determined.

Solving any system

  1. 1
    Row reduce the augmented matrix to row echelon form

    Use only the three elementary operations. Don't drop the augmented column.

  2. 2
    Scan for a row
    foundInconsistent — stop. The solution set is empty, and is not in the column space of .
    noneConsistent — continue.
  3. 3
    Count pivots: = rank. Compare with (the number of unknowns)
    r = nUnique solution. No free variables; back substitute.
    r < nInfinitely many, with free parameters.
  4. 4
    If there are free variables, finish with Gauss–Jordan

    Clear above each leading too, then set each free variable to a parameter and read the lead variables straight off.

  5. 5
    Write the answer in vector form

    Particular solution plus a combination of null-space vectors: . This form is what §3.2 and §3.6 build on.

Rank = number of nonzero rows in the echelon form of the coefficient matrix. This flow is the single most reusable procedure in the course.

Shape heuristics (and their exceptions)

ShapeUsual behaviourWhy it can fail
Overdetermined Usually inconsistentIf extra equations are redundant (dependent rows), it can be consistent — even with a unique solution.
Underdetermined Usually infinitely manyCan still be inconsistent. But it can never have exactly one solution: guarantees free variables.
Homogeneous Always consistent always works. The only question is whether there's anything else.
Theorem 1.2.1
An homogeneous system has a nontrivial solution whenever .

Proof in one line: at most pivots among columns leaves at least free variables, and each assignment of the free variables yields a solution.

Worked example

An underdetermined system with two free variables

0/5 steps
Solve using Gauss–Jordan reduction.
!Exam trap
Free variables are counted from the number of columns (unknowns), not rows: . A system with 6 equations, 4 unknowns and rank 4 has zero free variables no matter how tall it is.
Check your work

Row reduction / rank / null space

Exact fraction arithmetic — type entries like 3, -2, or 1/2. One row per line.

12-11
24-30
1215
→
1203
0012
0000
rank
2
nullity
2
cols (n)
4
pivot cols
1, 3
Basis for N(A)
-2
1
0
0
-3
0
-2
1
Basis for the column space — original columns 1, 3
1
2
1
-1
-3
1

Check yourself

Q1
The augmented matrix of a system reduces to the row echelon form below. What is the solution set?
T / F
An overdetermined system () is always inconsistent.
T / F
An underdetermined system () can never have exactly one solution.
Q4
A homogeneous system has equations in unknowns. What can you say for certain?