Rouché-Capelli theorem

According to the Rouché-Capelli theorem, the system is inconsistent if the rank of the augmented matrix is greater than the rank of the coefficient matrix.

He is most well known for Rouché's theorem in complex analysis, which he published in his alma mater's institutional journal in 1862, and for the Rouché-Capelli theorem in linear algebra.

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.

Specifically, according to the Rouché-Capelli theorem, any system of linear equations is inconsistent (has no solutions) if the rank of the augmented matrix is greater than the rank of the coefficient matrix; if, on the other hand, the ranks of these two matrices are equal, the system must have at least one solution.