Rouché–Capelli theorem is a theorem in linear algebra that determines the number of solutions for a system of linear equations, given the rank of its augmented matrix and coefficient matrix.
In particular: Consider the system of equations The coefficient matrix is and the augmented matrix is 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.
Indeed, an increase in the number of linearly independent columns has made the system of equations inconsistent.
The use of Gaussian elimination for putting the augmented matrix in reduced row echelon form does not change the set of solutions and the ranks of the involved matrices.
The theorem can be read almost directly on the reduced row echelon form as follows.