---
title: "Rouché-Capelli Theorem | Linear Algebra"
description: "Rouché-Capelli theorem tells you when a linear system is consistent by comparing coefficient and augmented matrix ranks in Linear Algebra and Differential Equations."
canonical: "https://fiveable.me/linear-algebra-and-differential-equations/key-terms/rouche-capelli-theorem"
type: "key-term"
subject: "Linear Algebra and Differential Equations"
unit: "Unit 10"
---

# Rouché-Capelli Theorem | Linear Algebra

## Definition

The Rouché-Capelli theorem says a linear system has at least one solution exactly when the coefficient matrix and augmented matrix have the same rank. In Linear Algebra and Differential Equations, that rank check tells you whether a system is consistent, unique, or infinite.

## What It Is

The Rouché-Capelli theorem is the rank test you use in Linear Algebra and Differential Equations to decide whether a linear system has solutions. It compares two matrices from the same system: the coefficient matrix, which holds the variable coefficients, and the augmented matrix, which adds the constants on the right side.

If the two ranks are equal, the system is consistent, meaning at least one solution exists. If the ranks are different, the system is inconsistent and no solution satisfies every equation at once. That one comparison saves you from trying to solve a system that cannot work.

The theorem also tells you how many solutions you get. When the common rank equals the number of variables, each variable gets pinned down and the system has one unique solution. When the common rank is smaller than the number of variables, some variables stay free, so you get infinitely many solutions.

A quick way to use the theorem is with Gaussian elimination. Row-reduce the augmented matrix, then compare the number of pivots in the coefficient part with the number of pivots after the constants are included. If a row turns into something like [0 0 0 | 5], that means a contradiction, so the ranks do not match and the system is inconsistent.

This theorem applies to both homogeneous and non-homogeneous systems. For a homogeneous system, the constant column is all zeros, so the ranks always match and you always have at least the trivial solution. The real question becomes whether you have only the trivial solution or also free-variable solutions, which depends on whether the rank reaches the number of variables.

## Why It Matters

Rouché-Capelli gives you a fast yes-or-no check before you spend time solving a linear system by substitution or elimination. In this course, that matters because many problems are not just asking for numbers, they are asking whether a model actually makes sense. If the system is inconsistent, you stop and explain why there is no solution instead of forcing algebra that cannot work.

It also connects directly to matrix ideas you use all semester. Rank tells you how much independent information is in the system, so the theorem ties together linear dependence, pivots, and solution behavior. That connection shows up again when you study differential equations systems, where matrix form is the starting point for analyzing behavior over time.

The theorem is especially useful when a problem asks about the number of solutions rather than the exact solution itself. You can tell the difference between one solution and infinitely many just by comparing rank to the number of variables. That is a lot faster than finishing a full elimination every time.

## Connections

### Rank

Rouché-Capelli is built on rank. You compare the rank of the coefficient matrix to the rank of the augmented matrix, and that comparison tells you whether the system is consistent. Rank is really the count of independent rows or columns, so it measures how much information the system actually contains.

### [Augmented Matrix](/linear-algebra-and-differential-equations/key-terms/augmented-matrix)

The augmented matrix is the setup you row-reduce when you apply the theorem. It combines the coefficients and constants into one array, which makes it easy to detect contradictions like a zero row ending with a nonzero constant. Without the augmented matrix, you cannot make the rank comparison the theorem needs.

### [Consistent System](/linear-algebra-and-differential-equations/key-terms/consistent-system)

The theorem gives the clean test for consistency. If the two ranks match, the system is consistent, so at least one solution exists. If they do not match, the system is inconsistent and has no solution, even if the equations look close to solvable at first glance.

### [inconsistent system](/linear-algebra-and-differential-equations/key-terms/inconsistent-system)

An inconsistent system is the failure case in Rouché-Capelli. After row reduction, it shows up as a contradiction, such as 0 = 5, in the augmented matrix. That contradiction means the augmented matrix has higher rank than the coefficient matrix, so the ranks are unequal.

## On the AP Exam

On a problem set or quiz, you usually use Rouché-Capelli after row-reducing a matrix. The task is to decide whether a system has no solution, one solution, or infinitely many solutions, not just to compute an answer mechanically. Look for pivots in the coefficient columns, then compare that count with the augmented matrix. If there is a contradictory row, name the system inconsistent. If the ranks match, use the number of variables to decide whether free variables remain. In differential equations work, this often shows up when you are checking a linear system before moving on to matrix methods or system analysis.

## rouché–capelli theorem vs Gaussian elimination

Gaussian elimination is the method you use to row-reduce a system, while Rouché-Capelli is the theorem you use to interpret the result. Elimination gets the matrix into a form where pivots and contradictions are visible. The theorem tells you what those pivots mean for consistency and the number of solutions.

## Key Takeaways

- Rouché-Capelli tells you whether a linear system has a solution by comparing the rank of the coefficient matrix with the rank of the augmented matrix.
- If the ranks are equal, the system is consistent. If the ranks are different, the system is inconsistent and has no solution.
- When the common rank equals the number of variables, the system has exactly one solution. When it is smaller, the system has infinitely many solutions.
- Row reduction is the practical way to apply the theorem, because it reveals pivots and contradictory rows quickly.
- For homogeneous systems, the ranks always match, so the real question is whether the trivial solution is the only solution or whether free variables appear.

## FAQs

### What is Rouché-Capelli theorem in Linear Algebra and Differential Equations?

It is the rank criterion for deciding whether a linear system has solutions. You compare the rank of the coefficient matrix to the rank of the augmented matrix. If they are equal, the system is consistent; if they are different, the system has no solution.

### How do you use Rouché-Capelli theorem?

Row-reduce the augmented matrix and count the pivots in the coefficient part versus the full augmented matrix. Matching ranks mean the system is solvable, and the number of variables then tells you whether the solution is unique or infinite. A contradictory row means the ranks differ.

### What is the difference between a consistent and inconsistent system here?

A consistent system has at least one solution, so the coefficient and augmented matrix ranks are the same. An inconsistent system has no solution because the augmented matrix introduces a contradiction that the coefficient matrix does not. That shows up as unequal ranks.

### Does the theorem work for homogeneous systems?

Yes. For a homogeneous system, the constant column is zero, so the ranks match automatically. That guarantees the trivial solution, and the rank compared with the number of variables tells you whether there are also nontrivial solutions.

## Related Study Guides

- [10.1 Linear Systems and Matrix Methods](/linear-algebra-and-differential-equations/unit-10/linear-systems-matrix-methods/study-guide/5rTXbMZFbYwoG7fI)

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