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Augmented matrix

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

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Definition

A simple definition follows.

Definition Let A be a $K	imes L$ matrix and $B$ a $K	imes M$ matrix. Then, the augmented matrix [eq1]is the [eq2] matrix obtained by appending the columns of $B$ to the right of those of A.

Note that the two matrices need to have the same number K of rows.

When we write the entries of the augmented matrix, a vertical line is used to visually separate the columns of A from those of $B$, as shown by the next example.

Example Let[eq3]Then, the augmented matrix [eq4] is[eq5]

The augmented matrix of a system of linear equations

Consider a linear system of K equations in $L$ unknowns [eq6]represented as[eq7]where A is the $K	imes L$ matrix of coefficients, x is the $L	imes 1$ vector of unknowns and $b$ is the Kx1 vector of constants.

The augmented matrix of the system is [eq8]

Example The system of two equations in three unknowns[eq9]can be represented in matrix form as[eq7]where[eq11]The augmented matrix of the system is[eq12]

Keeping track of elementary row operations

As we explained previously, there are two ways to perform an elementary row operation:

  1. perform the operation directly on the linear system;

  2. 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 matrix[eq13]where I is the identity matrix.

Example Consider the system of two equations in two unknowns[eq14]We have [eq15]Let us perform two row operations. In the first operation, we subtract $3$ times the first equation from the second, and we obtain[eq16]In the second operation, we multiply the second equation by $-1$:[eq17]From the augmented matrix, we can see that the transformed system is[eq18]and the identity matrix has been transformed into the matrix[eq19]We can easily verify that the original system pre-multiplied by $R$ is equal to the new system:[eq20]

Contemporaneous row operations on multiple systems

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 A.

Suppose you have two systems having the same coefficient matrix A but two different vectors of constants $b_{1}$ and $b_{2}$:[eq21]

Performing an elementary row operation on the augmented matrix[eq22]is the same as separately performing the operation on the two augmented matrices[eq23]and[eq24]

To understand why this is the case, denote by $R$ 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 $R$:[eq25]

Clearly, we can compute the same quantities by using the larger augmented matrix:[eq26]

Example Consider the system of two equations in two unknowns[eq27]together with a second system:[eq28]We have [eq29]Subtract the first row from the second:[eq30]Divide the second row by $2$:[eq31]Subtract the second row from the first:[eq32]Thus, the first system has become[eq33]and its solution is[eq34]The second system has become[eq35]and its solution is[eq36]

In the previous example, we transformed A into the identity matrix and $b_{1}$ and $b_{2}$ 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 into[eq37]then the first system is[eq38]and the second system is[eq39]

Using augmented matrices to derive inverse matrices

Given a full-rank $K	imes K$ matrix A, its inverse $A^{-1}$ satisfies the equation[eq40]where I is the $K	imes K$ identity matrix.

As a consequence, the problem of finding the inverse $A^{-1}$ is tantamount to finding the solutions of K systems of K equations in K unknowns:[eq41]where, in the k-th equation, the vector of unknowns [eq42] is the k-th column of $A^{-1}$ and the vector of constants $I_{ullet k}$ is the k-th column of I.

These K 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[eq43] to contemporaneously perform elementary row operations and solve the systems.

The augmented matrix can simply be written as[eq44]since [eq45] are the K columns of I.

If we transform the coefficient matrix A into the identity matrix by elementary row operations, then the columns of I are transformed into the solutions of the K systems of equations (as explained in the previous paragraph), and the augmented matrix becomes[eq46]

Example Define the matrix[eq47]Set up the augmented matrix[eq48]Divide the second row by $2$:[eq49]Subtract the second row from the first:[eq50]Then, the inverse of A is[eq51]

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