Page "Rank (linear algebra)" Paragraph 85

One useful application of calculating the rank of a matrix is the computation of the number of solutions of a system of linear equations.

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.

If, on the other hand, ranks of these two matrices are equal, the system must have at least one solution.

The solution is unique if and only if the rank equals the number of variables.

Otherwise the general solution has k free parameters where k is the difference between the number of variables and the rank.