Rouche-Capelli

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, Bosnia, Croatia, Romania, Serbia and Russia;

Rouché–Fontené theorem in France;

Rouché–Frobenius theorem in Spain and many countries in Latin America;

Frobenius theorem in Czechia and Slovakia.