In linear algebra, an augmented matrix matrix. This is usually done for the purpose of performing the same elementary row operations when solving a system of linear equations by Gaussian elimination.

linear equations depends only on the rank successive linear equations. According to the Rouché–Capelli theorem, any system of linear equations

-component column vector whose entries are the system's unknowns) is inconsistent (has no solutions) if the rank is greater than the rank of the coefficient matrix and the rank. In such a case there is an affine space of solutions of dimension equal to this difference.

identity matrix block to the identity matrix

Example of finding the inverse of a matrix

identity matrix to the identity matrix using elementary row operations.

Existence and number of solutions

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 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.

Solution of a linear system

As used in linear algebra, an augmented matrix is used to represent the coefficients

Note that the rank of the coefficient matrix, which is 3, equals the rank of the augmented matrix, so at least one solution exists; and since this rank equals the number of unknowns, there is exactly one solution.

so the solution of the system is .

References