Rouché–Capelli theorem

The Rouché–Capelli theorem is a theorem in linear algebra that allows computing the number of solutions in 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 Poland and Russia;
• Rouché–Capelli theorem in Italy;
• Rouché–Fontené theorem in France; and
• Rouché–Frobenius theorem in Spain and many countries in Latin America.

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 is an infinite number of solutions.

Example

Consider the system of equations

x + y + 2z = 3

x + y + z = 1

2x + 2y + 2z = 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 an infinite number of solutions.

In contrast, consider the system

x + y + 2z = 3

x + y + z = 1

2x + 2y + 2z = 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 rows has made the system of equations inconsistent.

See also

• Gaussian elimination

References

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