An augmented matrix is the result of joining the columns of two or more matrices having the same number of rows.
Augmented matrices are used in linear algebra to
parsimoniously represent systems of linear equations;
quickly perform and keep track of elementary row operations and transformations into equivalent systems;
contemporaneously perform elementary row operations on more than one system;
derive inverse matrices.
A simple definition follows.
Definition Let be a matrix and a matrix. Then, the augmented matrix is the matrix obtained by appending the columns of to the right of those of .
Note that the two matrices need to have the same number of rows.
When we write the entries of the augmented matrix, a vertical line is used to visually separate the columns of from those of , as shown by the next example.
Example LetThen, the augmented matrix is
Consider a linear system of equations in unknowns represented aswhere is the matrix of coefficients, is the vector of unknowns and is the vector of constants.
The augmented matrix of the system is
Example The system of two equations in three unknownscan be represented in matrix form aswhereThe augmented matrix of the system is
As we explained previously, there are two ways to perform an elementary row operation:
perform the operation directly on the linear system;
perform it on the identity matrix, and then pre-multiply the system by the transformed matrix you have obtained.
In case there are several elementary operations, we start from the identity matrix and sequentially perform all the operations on it. When we are done, we pre-multiply the original system by the matrix thus obtained. This procedure is equivalent to performing the sequence of operations directly on the system.
We can clearly see the two procedures in parallel by performing the sequence of desired operations on the augmented matrixwhere is the identity matrix.
Example Consider the system of two equations in two unknownsWe have Let us perform two row operations. In the first operation, we subtract times the first equation from the second, and we obtainIn the second operation, we multiply the second equation by :From the augmented matrix, we can see that the transformed system isand the identity matrix has been transformed into the matrixWe can easily verify that the original system pre-multiplied by is equal to the new system:
The augmented matrix can be used to contemporaneously perform elementary row operations on more than one system of equations, provided that all the systems have the same coefficient matrix .
Suppose you have two systems having the same coefficient matrix but two different vectors of constants and :
Performing an elementary row operation on the augmented matrixis the same as separately performing the operation on the two augmented matricesand
To understand why this is the case, denote by the matrix obtained by performing the operation on the identity matrix. Then, the operation can be performed on the two systems by pre-multiplying them by :
Clearly, we can compute the same quantities by using the larger augmented matrix:
Example Consider the system of two equations in two unknownstogether with a second system:We have Subtract the first row from the second:Divide the second row by :Subtract the second row from the first:Thus, the first system has becomeand its solution isThe second system has becomeand its solution is
In the previous example, we transformed into the identity matrix and and into the solutions of the two systems. This is a general result: when the coefficient matrix has been transformed into the identity matrix, then the vectors of constants have been transformed into the solutions of the linear systems. In fact, if the augmented matrix has been transformed intothen the first system isand the second system is
Given a full-rank matrix , its inverse satisfies the equationwhere is the identity matrix.
As a consequence, the problem of finding the inverse is tantamount to finding the solutions of systems of equations in unknowns:where, in the -th equation, the vector of unknowns is the -th column of and the vector of constants is the -th column of .
These systems have the same coefficient matrix. Therefore, we can use the technique illustrated in the previous section, that is, we can use the augmented matrix to contemporaneously perform elementary row operations and solve the systems.
The augmented matrix can simply be written assince are the columns of .
If we transform the coefficient matrix into the identity matrix by elementary row operations, then the columns of are transformed into the solutions of the systems of equations (as explained in the previous paragraph), and the augmented matrix becomes
Example Define the matrixSet up the augmented matrixDivide the second row by :Subtract the second row from the first:Then, the inverse of is
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