Rouché–Capelli theorem

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.

Not to be confused with Rouché's theorem.

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 ${\displaystyle \mathbb {R} ^{n}}$ 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 + 2z = 3,

x + y + z = 1,

2x + 2y + 2z = 2.

The coefficient matrix is

${\displaystyle A={\begin{bmatrix}1&1&2\\1&1&1\\2&2&2\\\end{bmatrix}},}$

and the augmented matrix is

${\displaystyle (A|B)=\left[{\begin{array}{ccc|c}1&1&2&3\\1&1&1&1\\2&2&2&2\end{array}}\right].}$

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 + 2z = 3,

x + y + z = 1,

2x + 2y + 2z = 5.

The coefficient matrix is

${\displaystyle A={\begin{bmatrix}1&1&2\\1&1&1\\2&2&2\\\end{bmatrix}},}$

and the augmented matrix is

${\displaystyle (A|B)=\left[{\begin{array}{ccc|c}1&1&2&3\\1&1&1&1\\2&2&2&5\end{array}}\right].}$

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

• Cramer's rule
• Gaussian elimination

References

1. 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