Rouché–Capelli theorem

Formal statement

A system of linear equations with n variables has a solution if and only if the rank of its coefficient matrix A is equal to the rank of its augmented matrix [A|b]. If there are solutions, they form an affine subspace of R n of dimension n − rank(A). In particular:

if n = rank(A), the solution is unique,

otherwise there is an infinite number of solutions.

Example

Consider the system of equations

x + y + 2z = 3 x + y + z = 1 2x + 2y + 2z = 2.

The coefficient matrix is

A = [ 1 1 2 1 1 1 2 2 2 ] ,

and the augmented matrix is

( A | B ) = [ 1 1 2 3 1 1 1 1 2 2 2 2 ] .

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 an infinite number of solutions.

In contrast, consider the system

x + y + 2z = 3 x + y + z = 1 2x + 2y + 2z = 5.

The coefficient matrix is

A = [ 1 1 2 1 1 1 2 2 2 ] ,

and the augmented matrix is

( A | B ) = [ 1 1 2 3 1 1 1 1 2 2 2 5 ] .

In this example the coefficient matrix has rank 2 while the augmented matrix has rank 3; so this system of equations has no solution. Indeed, an increase in the number of linearly independent rows has made the system of equations inconsistent.