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:
Kronecker–Capelli theorem in Austria, Poland, Romania and Russia;
Rouché–Capelli theorem in Italy;
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 \( \mathbb {R} ^{n} \) 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
\( A={\begin{bmatrix}1&1&2\\1&1&1\\2&2&2\\\end{bmatrix}}, \)
and the augmented matrix is
\( (A|B)=\left[{\begin{array}{ccc|c}1&1&2&3\\1&1&1&1\\2&2&2&2\end{array}}\right]. \)
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
\( A={\begin{bmatrix}1&1&2\\1&1&1\\2&2&2\\\end{bmatrix}}, \)
and the augmented matrix is
\( (A|B)=\left[{\begin{array}{ccc|c}1&1&2&3\\1&1&1&1\\2&2&2&5\end{array}}\right]. \)
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
Cramer's rule
Gaussian elimination
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.
External links
Kronecker-Capelli Theorem at Wikibooks
Kronecker-Capelli's Theorem - youtube video with a proof
Kronecker-Capelli theorem in the Encyclopaedia of Mathematics
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
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