Rouché–Capelli theorem

Rouché–Capelli theorem is the theorem in linear algebra that allows computing the number of solutions in a system of linear equations given the ranks of its augmented matrix and coefficient matrix. The theorem is known as Kronecker–Capelli theorem in Russia, Rouché–Capelli theorem in Italy, Rouché–Fontené theorem in France and Rouché–Frobenius theorem in Spain and many countries in Latin America.

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]. 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 infinite number of solutions.

References

• A. Carpinteri (1997). Structural mechanics. Taylor and Francis. p. 74. ISBN 0-419-19160-7.