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Solution Types

Gaussian elimination doesn't just find solutions — it tells you exactly what kind of solution a system has: unique, none, or infinitely many. The key lies in the rank of the coefficient matrix.

~12 min read M3 · L3 Intermediate

Three Possible Outcomes

After Gaussian elimination transforms a linear system into Row Echelon Form, the RREF tells a complete story about the nature of the solution set. Every linear system Ax = b falls into exactly one of three categories:

Unique Solution
Exactly one x
Every variable is a pivot variable. The RREF is [I|x*]. The system is consistent and the solution is determined.
No Solution
Inconsistent
A contradiction row appears: all zeros on the left, nonzero on the right. The planes described by the equations never share a common point.
Infinitely Many
Parametric family
At least one free variable exists. The solution set is a line, plane, or higher-dimensional subspace parameterized by the free variables.

The decision between these three cases is determined entirely by comparing two numbers: the rank of the coefficient matrix and the rank of the augmented matrix. Understanding rank — and how to read it from the RREF — is the central skill of this lesson.

Rank: The Most Important Number

The rank of a matrix A, written rank(A), is the number of pivot positions in its Row Echelon Form. It counts how many independent constraints the system actually imposes — equivalently, the number of dimensions of information the matrix contains.

For a system of m equations in n unknowns, the rank satisfies rank(A) ≤ min(m, n). Columns without pivot positions correspond to free variables — unknowns that can take any value, with all other variables then determined as functions of them. The number of free variables is n − rank(A).

Rank–Nullity Theorem
\text{rank}(A) + \text{nullity}(A) = n
For any m×n matrix A, the number of pivot variables (rank) plus the number of free variables (nullity) always equals n, the total number of columns. This identity is a cornerstone of linear algebra — it quantifies exactly how much freedom remains after the constraints are imposed.

The Consistency Condition

The augmented matrix [A|b] has one extra column — the right-hand side. Its rank can be either equal to rank(A) or one greater. This difference determines whether the system is consistent.

The Fundamental Consistency Test

A system Ax = b is consistent (has at least one solution) if and only if rank(A) = rank([A|b]). In other words: the right-hand side b must not introduce a new pivot position. If it does, the augmented matrix has a row of the form [0 0 … 0 | c] with c ≠ 0, which says "0 = c" — an impossibility.

Geometrically, the rows of A describe hyperplanes in n-dimensional space. If the system is inconsistent, those hyperplanes have no common intersection point — the right-hand side b lies outside the column space of A.

Inconsistent System — Contradiction Row
\left[\begin{array}{ccc|c}1&2&-1&3\\0&1&4&7\\0&0&0&5\end{array}\right]
The last row reads 0·x₁ + 0·x₂ + 0·x₃ = 5, which is impossible. No matter what values x₁, x₂, x₃ take, the left side is zero but the right side is 5. The system has no solution.

Unique Solution: Full Rank Square Systems

A square n×n system has a unique solution exactly when rank(A) = n — that is, when every column is a pivot column and no free variables exist. In this case the RREF of the augmented matrix is [I|x*], where I is the n×n identity and x* is the unique solution vector.

This situation corresponds to the coefficient matrix A being invertible. When A is invertible, the unique solution is x* = A⁻¹b. The determinant of A is nonzero, and the columns of A form a basis for Rⁿ.

Unique Solution — RREF Form
\left[\begin{array}{ccc|c}1&0&0&2\\0&1&0&-1\\0&0&1&4\end{array}\right]
The RREF shows every variable determined: x₁ = 2, x₂ = −1, x₃ = 4. Three pivots for three variables — no free variables, no ambiguity. This system has exactly one solution.

Infinitely Many Solutions: Free Variables

When rank(A) = rank([A|b]) but rank(A) < n, the system is consistent but underdetermined. There are n − rank(A) free variables, and the solution set is an affine subspace of Rⁿ with that many dimensions.

The solution is written as a parametric form: one particular solution x_p plus any element of the null space of A (the set of all x satisfying Ax = 0). Free variables act as parameters — choose any value for them, and the pivot variables are determined by the RREF equations.

System with One Free Variable
\left[\begin{array}{ccc|c}1&0&2&5\\0&1&-3&1\end{array}\right] \Rightarrow \begin{cases}x_1 = 5 - 2t \\ x_2 = 1 + 3t \\ x_3 = t\end{cases}
With two equations in three unknowns, x₃ is free (no pivot in column 3). Setting x₃ = t, the pivot variables are determined: x₁ and x₂ become functions of t. The solution set is a line in R³, parameterized by t ∈ R.

Reading the Parametric Solution

From the RREF, each pivot row gives one pivot variable in terms of the free variables. The procedure is:

  1. Identify all free variable columns (non-pivot columns).
  2. Assign a parameter (t, s, …) to each free variable.
  3. Express each pivot variable using the corresponding RREF row, substituting the parameter values.
  4. Write the solution as x = x_p + t·v₁ + s·v₂ + … where x_p is any particular solution and v₁, v₂, … span the null space.
Particular Solution + Null Space

The complete solution to Ax = b (when consistent) is always x = x_p + x_h, where x_p is any particular solution (set all free variables to 0 to get a convenient one) and x_h is the general solution to the homogeneous system Ax = 0. This additive structure is a deep property of linear systems — it parallels how differential equations have particular plus homogeneous solutions.

The Three-Case Decision Tree

After row-reducing [A|b] to REF or RREF, the classification is immediate. Let r = rank(A) and r̃ = rank([A|b]):

r̃ > r
No solution
A contradiction row exists. b ∉ col(A). The system is inconsistent. The planes have no common point.
r̃ = r = n
Unique solution
Consistent, no free variables. Every column has a pivot. A is square and invertible (if m = n).
r̃ = r < n
Infinitely many
Consistent, with n − r free variables. The solution set is an (n − r)-dimensional affine subspace.

Homogeneous Systems Always Have a Solution

The homogeneous system Ax = 0 (right-hand side = 0) is always consistent — x = 0 is always a solution, called the trivial solution. The question is whether there are additional, nontrivial solutions. The answer: a nontrivial solution exists if and only if rank(A) < n, i.e., there is at least one free variable.

The complete solution set of Ax = 0 forms a subspace of Rⁿ — the null space (or kernel) of A. Its dimension is n − rank(A), a quantity called the nullity of A. The null space captures exactly the directions in which A "collapses" information — vectors that become 0 after multiplication by A.

Null Space Definition
\text{null}(A) = \{\mathbf{x} \in \mathbb{R}^n : A\mathbf{x} = \mathbf{0}\}
The null space of A is the set of all vectors x that A maps to zero. It is always a subspace — closed under addition and scalar multiplication. Its dimension (the nullity) equals the number of free variables in the homogeneous system Ax = 0.

Worked Example: All Three Cases

Case 1 — Unique Solution

System: x₁ + 2x₂ = 5, 3x₁ − x₂ = 1. The augmented matrix row-reduces to [I|x*] with pivots in both columns. rank(A) = 2 = n, so the unique solution is x₁ = 1, x₂ = 2.

Case 2 — No Solution

System: x₁ + x₂ = 3, 2x₁ + 2x₂ = 7. The second equation is twice the first — but the right-hand side is 7, not 6. After row reduction: R₂ ← R₂ − 2R₁ gives [0 0 | 1], a contradiction. The system is inconsistent: the two lines in the plane are parallel and never intersect.

Case 3 — Infinitely Many Solutions

System: x₁ + x₂ + x₃ = 3, 2x₁ + 2x₂ + 2x₃ = 6. The second equation is exactly twice the first. After row reduction, only one nonzero row remains: [1 1 1 | 3]. rank(A) = 1, n = 3, so there are 2 free variables. Setting x₂ = s and x₃ = t, the solution is x₁ = 3 − s − t — a 2-dimensional plane in R³.

Geometric Intuition

In R³, three linear equations each describe a plane. The solution set is the intersection of these planes: a point (unique), the empty set (no solution), a line (one free variable), or a plane (two free variables, meaning the planes all coincide). The rank tells you the dimension of the "information" the equations collectively provide — and the geometry of the intersection follows directly.

Key Takeaways

Every linear system has exactly one of three solution types. Inconsistency occurs when b lies outside the column space of A — detectable as a contradiction row in the RREF. A unique solution requires rank(A) = n, meaning every variable is pinned by a pivot. When rank(A) < n and the system is consistent, infinitely many solutions form an affine subspace parameterized by n − rank(A) free variables. The complete solution is always particular plus homogeneous: x = x_p + x_h. The null space of A — the set of all solutions to Ax = 0 — encodes exactly how much freedom the system leaves, and its dimension is the nullity n − rank(A).