Rouché–Capelli theorem

Rouché–Capelli theorem is a theorem in linear algebra that determines the number of solutions of a system of linear equations, given the ranks 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, Ukraine, 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.