Rouché–Capelli theorem
In linear algebra, the Rouché–Capelli theorem 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:
- Rouché–Capelli theorem in English speaking countries, Italy and Brazil ;
- Kronecker–Capelli theorem in Austria, Poland , Croatia, Romania, Serbia and Russia ;
- 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.
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 [math]\displaystyle{ \mathbb{R}^n }[/math] 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
- [math]\displaystyle{ A = \begin{bmatrix} 1 & 1 & 2 \\ 1 & 1 & 1 \\ 2 & 2 & 2 \\ \end{bmatrix}, }[/math]
and the augmented matrix is
- [math]\displaystyle{ (A|B) = \left[\begin{array}{ccc|c} 1 & 1 & 2 & 3\\ 1 & 1 & 1 & 1 \\ 2 & 2 & 2 & 2 \end{array}\right]. }[/math]
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
- [math]\displaystyle{ A = \begin{bmatrix} 1 & 1 & 2 \\ 1 & 1 & 1 \\ 2 & 2 & 2 \\ \end{bmatrix}, }[/math]
and the augmented matrix is
- [math]\displaystyle{ (A|B) = \left[\begin{array}{ccc|c} 1 & 1 & 2 & 3\\ 1 & 1 & 1 & 1 \\ 2 & 2 & 2 & 5 \end{array}\right]. }[/math]
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
References
- ↑ Shafarevich, Igor R.; Remizov, Alexey (2012-08-23) (in en). Linear Algebra and Geometry. Springer Science & Business Media. p. 56. ISBN 9783642309946. https://books.google.com/books?id=6Pp2-DTOKWIC&pg=PA56.
- 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
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