# Rouché–Capelli theorem

Rouché–Capelli theorem Not to be confused with Rouché's 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, Romania 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. Contents 1 Formal statement 2 Example 3 See also 4 References 5 External links 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 {displaystyle 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 {displaystyle A={begin{bmatrix}1&1&2\1&1&1\2&2&2\end{bmatrix}},} and the augmented matrix is {displaystyle (A|B)=left[{begin{array}{ccc|c}1&1&2&3\1&1&1&1\2&2&2&2end{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 {displaystyle A={begin{bmatrix}1&1&2\1&1&1\2&2&2\end{bmatrix}},} and the augmented matrix is {displaystyle (A|B)=left[{begin{array}{ccc|c}1&1&2&3\1&1&1&1\2&2&2&5end{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 Categories: Theorems in linear algebraMatrix theory

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