In linear algebra, an augmented matrix
matrix obtained by appending a
This is usually done for the purpose of performing the same elementary row operations on the augmented matrix
when solving a system of linear equations by Gaussian elimination.
of unknowns, the number of solutions to a system of
linear equations depends only on the rank of the matrix of coefficients
representing the system and the rank of the corresponding augmented matrix
consist of the right hand sides of the
According to the Rouché–Capelli theorem, any system of linear equations where
-component column vector whose entries are the unknowns of the system is inconsistent (has no solutions) if the rank of the augmented matrix
is greater than the rank of the coefficient matrix
If, on the other hand, the 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
is the difference between the number of variables
In such a case there as an affine space of solutions of dimension equal to this difference.
The inverse of a nonsingular square matrix
dimensional augmented matrix
Applying elementary row operations to transform the left-hand
block to the identity matrix
block is then the inverse matrix
we form the augmented matrix
to the identity matrix using elementary row operations on
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.
As used in linear algebra, an augmented matrix is used to represent the coefficients and the solution vector of each equation set.
the coefficients and constant terms give the matrices
and hence give the augmented matrix
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.
To obtain the solution, row operations can be performed on the augmented matrix to obtain the identity matrix on the left side, yielding