In linear algebra, an augmented matrix is a matrix obtained by appending a -dimensional row vector , on the right, as a further column to a -dimensional matrix . This is usually done for the purpose of performing the same elementary row operations on the augmented matrix as is done on the original one when solving a system of linear equations by Gaussian elimination.
For example, given the matrices and column vector , where
For a given number 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 where the components of consist of the right hand sides of the successive linear equations. According to the Rouché–Capelli theorem, any system of linear equations
where is the -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 . Otherwise the general solution has free parameters where is the difference between the number of variables and the rank. In such a case there as an affine space of solutions of dimension equal to this difference.
The inverse of a nonsingular square matrix of dimension may be found by appending the identity matrix to the right of to form the dimensional augmented matrix . Applying elementary row operations to transform the left-hand block to the identity matrix , the right-hand block is then the inverse matrix
Let be the square 2×2 matrix
To find the inverse of we form the augmented matrix where is the identity matrix. We then reduce the part of corresponding to to the identity matrix using elementary row operations on .
Consider the system of equations
The coefficient 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
The coefficient 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.
As used in linear algebra, an augmented matrix is used to represent the coefficients and the solution vector of each equation set. For the set of equations
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