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:
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 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
![(A|B)=\left[{\begin{array}{ccc|c}1&1&2&3\\1&1&1&1\\2&2&2&2\end{array}}\right].](../I/m/719fe63e6ea283f189df8363c1d418f0a73a5ebb.svg)
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
![(A|B)=\left[{\begin{array}{ccc|c}1&1&2&3\\1&1&1&1\\2&2&2&5\end{array}}\right].](../I/m/d8d486765032f65d6f336123d4e956685ad6257d.svg)
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
- ↑ 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.