Rouché–Capelli theorem

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:

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:

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

References

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