Rouché–Capelli theorem

A system of linear equations with variables has a solution if and only if the the rank of its coefficient matrix is equal to the rank of its augmented matrix . In particular:

• if , the solution is unique
• otherwise there are infinitely many solutions.

Example 1: System of equations:

Coefficient matrix:

Augmented matrix:

Since both of these have the same rank, namely , there exists at least one solution; and since their rank is less than the number of unknowns, the latter being , there are infinitely many solutions.

Example 2: System of equations:

Coefficient matrix:

Augmented matrix:

In this example the coefficient matrix has rank , while the augmented matrix has rank ; so this system of equations has no solution. Indeed, an increase in the number of linearly independent columns has made the system of equations inconsistent.