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Solvability of Linear Systems



The Two Questions
The Existence Condition
The Uniqueness Condition
The Three Cases Combined
The Rouché–Capelli Theorem
Square Systems
Overdetermined Systems
Underdetermined Systems
Geometric Interpretation
Structure of the Solution Set
Summary: Solvability Questions Answered



When Solutions Exist and When They Are Unique

Two separate questions govern every linear system: does at least one solution exist, and if so, is that solution the only one? The rank of the coefficient matrix answers both. A single integer determines whether the system is inconsistent, uniquely solvable, or infinitely underdetermined — and characterizes the geometry of the solution set in each case.



The Two Questions

Given the system Ax=bA\mathbf{x} = \mathbf{b} with AA of size m×nm \times n, two logically independent questions arise.

Existence: is there at least one vector x\mathbf{x} satisfying Ax=bA\mathbf{x} = \mathbf{b}? This asks whether b\mathbf{b} lies in the column space of AA.

Uniqueness: if a solution exists, is it the only one? This asks whether the null space of AA is trivial.

The two questions are independent — existence can hold without uniqueness, and non-existence makes uniqueness moot. Both are answered by the rank of AA.

The Existence Condition

The system Ax=bA\mathbf{x} = \mathbf{b} has at least one solution if and only if

rank(A)=rank([A∣b])\text{rank}(A) = \text{rank}([A \mid \mathbf{b}])


When this condition holds, b\mathbf{b} is already a linear combination of the columns of AA, so appending b\mathbf{b} as an extra column does not introduce a new independent direction — the rank stays the same.

When the condition fails — rank([A∣b])>rank(A)\text{rank}([A \mid \mathbf{b}]) > \text{rank}(A) — the vector b\mathbf{b} is not in the column space. In the echelon form of the augmented matrix, this appears as a row [0  0  ⋯  0∣d][0 \; 0 \; \cdots \; 0 \mid d] with d≠0d \neq 0: the equation 0=d0 = d has no solution, and the system is inconsistent.

Since appending one column can increase the rank by at most 11, the only possibilities are rank([A∣b])=rank(A)\text{rank}([A \mid \mathbf{b}]) = \text{rank}(A) (consistent) or rank([A∣b])=rank(A)+1\text{rank}([A \mid \mathbf{b}]) = \text{rank}(A) + 1 (inconsistent).

The Uniqueness Condition

When a solution exists, it is unique if and only if

rank(A)=n\text{rank}(A) = n


where nn is the number of unknowns. Full column rank means every column of AA contains a pivot, leaving no free variables. The null space is {0}\{\mathbf{0}\}, so the particular solution xp\mathbf{x}_p stands alone with nothing to add.

When rank(A)<n\text{rank}(A) < n, there are n−rank(A)n - \text{rank}(A) free variables. Each free variable parametrizes a direction along which the solution can move without violating the equations. The solution set is infinite — a translated copy of the null space, which has dimension n−rank(A)n - \text{rank}(A).

The Three Cases Combined

Putting existence and uniqueness together, every linear system falls into exactly one of three cases.

rank(A)<rank([A∣b])\text{rank}(A) < \text{rank}([A \mid \mathbf{b}]): no solution. The system is inconsistent.

rank(A)=rank([A∣b])=n\text{rank}(A) = \text{rank}([A \mid \mathbf{b}]) = n: exactly one solution. The system is consistent and fully determined.

rank(A)=rank([A∣b])<n\text{rank}(A) = \text{rank}([A \mid \mathbf{b}]) < n: infinitely many solutions. The system is consistent but underdetermined, with n−rank(A)n - \text{rank}(A) free parameters.

There is no case with a finite number of solutions greater than one. If two distinct solutions exist, their difference lies in the null space, which is a subspace — and a nontrivial subspace contains infinitely many vectors, generating infinitely many solutions.

The Rouché–Capelli Theorem

The existence condition rank(A)=rank([A∣b])\text{rank}(A) = \text{rank}([A \mid \mathbf{b}]) is known as the Rouché–Capelli theorem (or the Kronecker–Capelli theorem in some traditions). It states:

The system Ax=bA\mathbf{x} = \mathbf{b} is consistent if and only if the coefficient matrix and the augmented matrix have the same rank. When consistent, the solution set has dimension n−rank(A)n - \text{rank}(A).

The theorem unifies the existence and dimension questions into a single rank comparison. It applies to every linear system regardless of the shape of AA — square, tall, or wide. The proof follows directly from the column-space interpretation: b∈Col(A)\mathbf{b} \in \text{Col}(A) if and only if adding b\mathbf{b} as a column does not increase the rank.

Square Systems

When AA is n×nn \times n, the determinant provides the sharpest diagnostic.

If det⁡(A)≠0\det(A) \neq 0: the matrix is invertible, the rank is nn, and the system Ax=bA\mathbf{x} = \mathbf{b} has exactly one solution for every b\mathbf{b}. The solution is x=A−1b\mathbf{x} = A^{-1}\mathbf{b}, or equivalently, the solution obtained by Gaussian elimination. Existence and uniqueness both hold universally — the right-hand side does not matter.

If det⁡(A)=0\det(A) = 0: the matrix is singular, the rank is less than nn, and the outcome depends on b\mathbf{b}. For some b\mathbf{b} (those in the column space), infinitely many solutions exist. For other b\mathbf{b} (those outside the column space), no solution exists. The determinant test determines whether the coefficient matrix is adequate; the rank comparison with the augmented matrix determines whether the specific b\mathbf{b} is reachable.

Overdetermined Systems

When m>nm > n — more equations than unknowns — the system is overdetermined. The coefficient matrix is tall, and generically, no solution exists.

The column space of AA is at most nn-dimensional inside Rm\mathbb{R}^m. When m>nm > n, this column space is a proper subspace — most vectors in Rm\mathbb{R}^m lie outside it. A randomly chosen b\mathbf{b} will almost certainly not be in the column space, making the system inconsistent.

A solution exists only when b\mathbf{b} happens to lie in the column space — when the extra equations are consistent with the first nn. When a solution does exist and rank(A)=n\text{rank}(A) = n, it is unique.

When no exact solution exists, the least-squares approach finds the x^\hat{\mathbf{x}} that minimizes ∥Ax−b∥2\|A\mathbf{x} - \mathbf{b}\|^2 — the closest approximation. The least-squares solution satisfies the normal equations ATAx^=ATbA^T A \hat{\mathbf{x}} = A^T \mathbf{b} and is the projection of b\mathbf{b} onto the column space.

Underdetermined Systems

When m<nm < n — fewer equations than unknowns — the system is underdetermined. If a solution exists, it is never unique: the rank cannot exceed mm, which is less than nn, so at least n−mn - m free variables remain.

The solution set, when nonempty, is an affine subspace of dimension at least n−mn - m. Multiple vectors x\mathbf{x} satisfy the equations, and additional criteria beyond the linear system itself are needed to select a preferred one.

Common selection criteria include minimum norm (the x\mathbf{x} closest to 0\mathbf{0}, given by the pseudoinverse x=AT(AAT)−1b\mathbf{x} = A^T(AA^T)^{-1}\mathbf{b}), sparsity (the x\mathbf{x} with the fewest nonzero entries, central to compressed sensing), and physical constraints (bounds or nonnegativity in engineering applications).

Underdetermined systems are not defective — they arise naturally whenever a problem has more degrees of freedom than constraints. The linear system identifies the feasible set; the selection criterion picks a point within it.
Shape Distinguishing condition Existence for a given b Uniqueness Practical approach
Square (m = n), det(A) ≠ 0 rank(A) = n; A invertible always exists for every b always unique x = A⁻¹b; or Gaussian elimination
Square (m = n), det(A) = 0 rank(A) < n; A singular depends on b — exists iff b ∈ Col(A) never unique (when it exists) row reduce [A | b]; compare ranks
Overdetermined (m > n) A is tall; column space is a proper subspace of ℝᵐ unusual — most b fall outside Col(A) unique when it exists and rank(A) = n when inconsistent: least squares via AᵀA x̂ = Aᵀb
Underdetermined (m < n) A is wide; rank ≤ m < n forces free variables depends on b — exists iff rank(A) = rank([A | b]) never unique; at least n − m free parameters minimum-norm via pseudoinverse Aᵀ(AAᵀ)⁻¹b, or other selection criterion

Geometric Interpretation

Each equation ai1x1+ai2x2+⋯+ainxn=bia_{i1}x_1 + a_{i2}x_2 + \cdots + a_{in}x_n = b_i defines a hyperplane in Rn\mathbb{R}^n — a flat set of dimension n−1n - 1. The solution set of the full system is the intersection of all mm hyperplanes.

When the system is inconsistent, the hyperplanes have no common point. In R2\mathbb{R}^2 this means parallel lines. In R3\mathbb{R}^3 it can mean parallel planes, or planes arranged in a triangular prism where each pair intersects but no point lies on all three.

When the system has a unique solution, the hyperplanes meet at a single point. This requires at least nn independent equations — enough to cut the solution space down from nn dimensions to 00.

When the system has infinitely many solutions, the hyperplanes meet along a flat of dimension n−rn - r, where r=rank(A)r = \text{rank}(A). Each independent equation reduces the dimension of the intersection by one, and the rank counts how many equations cut independently. The remaining n−rn - r dimensions are the free directions in the solution set.

Structure of the Solution Set

When Ax=bA\mathbf{x} = \mathbf{b} is consistent, the complete solution set has the form

{xp+xh:xh∈Null(A)}\{\mathbf{x}_p + \mathbf{x}_h : \mathbf{x}_h \in \text{Null}(A)\}


where xp\mathbf{x}_p is any one particular solution. This set is a coset of the null space — the null space translated by xp\mathbf{x}_p. In geometric terms, it is an affine subspace: a subspace shifted away from the origin.

The dimension of the solution set equals the dimension of the null space: n−rank(A)n - \text{rank}(A). When this is 00, the solution set is a single point {xp}\{\mathbf{x}_p\}. When it is 11, the solution set is a line. When it is 22, a plane. The shape is always a flat, and the null space determines its orientation.

The particular solution xp\mathbf{x}_p captures the effect of the right-hand side b\mathbf{b}. The homogeneous component xh\mathbf{x}_h captures the inherent freedom in the system — the directions along which the solution can shift without violating any equation. This decomposition into "forced" and "free" parts is one of the most fundamental structural results in linear algebra.
Component Symbol Where it comes from What it captures
Particular solution xp any one vector satisfying A xp = b the "forced" part; the response to the right-hand side b
Homogeneous part xh any vector in Null(A), i.e. A xh = 0 the "free" directions along which the solution can shift
Complete solution x = xp + xh sum over all xh ∈ Null(A) affine flat of dimension n − rank(A) through xp

Summary: Solvability Questions Answered

The two foundational questions about A·x = b — existence and uniqueness — together with the structure of the solution set and the special behavior of square, overdetermined, and underdetermined shapes, can be collected as a six-row Q&A reference. The table below pairs each question a reader might bring to a linear system with the answer in terms of rank, determinant, or solution-set structure.
Question about A x = b Answer
Does a solution exist? yes ⟺ rank(A) = rank([A | b]); equivalently, b ∈ Col(A) (Rouché–Capelli)
If so, is the solution unique? yes ⟺ rank(A) = n; equivalently, Null(A) = {0} (no free variables)
What does the solution set look like? {xp + xh : xh ∈ Null(A)} — affine flat of dimension n − rank(A)
A is square with det(A) ≠ 0? unique solution for every b: x = A⁻¹b
Overdetermined (m > n) and no exact solution? use least squares: x̂ minimizes ‖Ax − b‖² via AᵀA x̂ = Aᵀb
Underdetermined (m < n), need a specific x? apply a selection criterion: minimum norm via the pseudoinverse, or sparsity, or physical constraints