The Rouché–Capelli theorem is a theorem in linear algebra that allows computing the number of solutions in a system of linear equations given the ranks of its augmented matrix and coefficient matrix. The theorem is also known as the:
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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
Example
Consider the system of equations
x + y + 2z = 3x + y + z = 12x + 2y + 2z = 2.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 an infinite number of solutions.
In contrast, consider the system
x + y + 2z = 3x + y + z = 12x + 2y + 2z = 5.The coefficient matrix is
and the augmented matrix is
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.