The **Rouché–Capelli theorem** is a theorem in linear algebra that determines the number of solutions for a system of linear equations, given the rank of its augmented matrix and coefficient matrix. The theorem is variously known as the:

**Kronecker–Capelli theorem**in Austria, Poland, Romania and Russia;**Rouché–Capelli theorem**in Italy;**Rouché–Fontené theorem**in France;**Rouché–Frobenius theorem**in Spain and many countries in Latin America;**Frobenius theorem**in the Czech Republic and in Slovakia.

## Formal statement

A system of linear equations with *n* variables has a solution if and only if the rank of its coefficient matrix *A* is equal to the rank of its augmented matrix [*A*|*b*].^{[1]} If there are solutions, they form an affine subspace of of dimension *n* − rank(*A*). In particular:

- if
*n*= rank(*A*), the solution is unique, - otherwise there are infinitely many solutions.

## Example

Consider the system of equations

*x*+*y*+ 2*z*= 3,*x*+*y*+*z*= 1,- 2
*x*+ 2*y*+ 2*z*= 2.

The coefficient matrix is

and the augmented matrix is

Since both of these have the same rank, namely 2, there exists at least one solution; and since their rank is less than the number of unknowns, the latter being 3, there are infinitely many solutions.

In contrast, consider the system

*x*+*y*+ 2*z*= 3,*x*+*y*+*z*= 1,- 2
*x*+ 2*y*+ 2*z*= 5.

The coefficient matrix is

and the augmented matrix is

In this example the coefficient matrix has rank 2, while the augmented matrix has rank 3; so this system of equations has no solution. Indeed, an increase in the number of linearly independent columns has made the system of equations **inconsistent**.

## See also

## References

**^**Shafarevich, Igor R.; Remizov, Alexey (2012-08-23).*Linear Algebra and Geometry*. Springer Science & Business Media. p. 56. ISBN 9783642309946.

- A. Carpinteri (1997).
*Structural mechanics*. Taylor and Francis. p. 74. ISBN 0-419-19160-7.

## External links

- Kronecker-Capelli Theorem at Wikibooks
- Kronecker-Capelli's Theorem - youtube video with a proof
- Kronecker-Capelli theorem in the Encyclopaedia of Mathematics