Rouché–Capelli theorem

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The RouchéCapelli 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[edit]

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[edit]

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[edit]

References[edit]

  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.