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Mathematical Expression Editor

We introduce the augmented matrix notation and solve linear system by carrying
augmented matrices to row-echelon or reduced row-echelon form.

SYS-0020: Augmented Matrix Notation and Elementary Row Operations

Augmented Matrix Notation

Recall that the following three operations performed on a linear system are called
elementary row operations

Switching the order of two equations

Multiplying both sides of an equation by the same non-zero constant

Adding a multiple of one equation to another

In SYS-0010 we discussed why applying elementary row operations to a linear system
results in an equivalent system - a linear system with the same solution
set.

In this module we seek an efficient method for recording our computations and
arriving at a solution using elementary row operations.

Consider the linear system
Our goal is to use elementary row operations to transform this system into an
equivalent system of the form
We have to keep in mind that given an arbitrary system, an equivalent system of this
form may not exist (we will talk a lot more about this later), but it does exist in this
case, and we would like to find a more efficient way of finding it than having to write
and rewrite our equations at each step.

In this problem, we prompt you to perform elementary row operations and ask you to
fill in the coefficients in the resulting equations.

If we drop all of the zero terms, we have:
Now we see that is the solution.

Observe that throughout the entire process, variables , , and remained in place; only
the coefficients in front of the variables and the entries on the right changed. Let’s try
to recreate this process without writing down the variables. We can capture the
original system in (eq:sys20originalsystem1) as follows:

This array is called an augmented matrix. The side to the left of the vertical bar is
called the coefficient matrix, while the side to the right of the vertical bar
corresponds to the constants on the right side of the system.

We can capture all of the elementary row operations we performed earlier as
follows:

The last augmented matrix corresponds to systems in (eq:sys20rref1) and (eq:sys20rrefnozeros), and we can easily see
the solution.

Exploration init:augmentedmatrixex introduced us to some vocabulary terms. Let’s formalize our definitions.
Every linear system
can be written in the augmented matrix form as follows:
The array to the left of the vertical bar is called the coefficient matrix of the linear
system and is often given a capital letter name, like . The vertical array to the right
of the bar is called a constant vector.
We will sometimes use the following notation to represent an augmented
matrix.

Consider the system
Recall that in Exploration init:augmentedmatrixex we first converted the given system to an augmented
matrix form, then performed elementary row operations until we arrived at a
“convenient” form. We then converted the “convenient” augmented matrix back to a
system of equations and identified the solution. The term “convenient” is open to
interpretation. In this problem we will explore two “convenient” forms. Each one will
lead to a definition.

The augmented matrix in (eq:sys20reducedrowechelon) has the same convenient form as the one in (eq:sys20rref). This
augmented matrix in corresponds to the system

This gives us the solution .

While the augmented matrix in (eq:sys20reducedrowechelon) was certainly “convenient”, we could have
converted back to the equation format a little earlier. Let’s take a look at the
augmented matrix in (eq:sys20rowechelon). Converting (eq:sys20rowechelon) to a system of equations gives us

Substituting into the second equation and solving for gives us
Now substituting and into the first equation results in
This process is called back substitution and it produces the same solution as we
obtained earlier.

Observe that the coefficient matrices in (eq:sys20rref) and (eq:sys20reducedrowechelon) have the same format: 1’s along the
diagonal, zeros above and below the 1’s. The other “convenient” format, exhibited by
the coefficient matrix in (eq:sys20rowechelon), also has zeros below the diagonal, but not all of the
diagonal entries are 1’s and some of the entries above the diagonal are not zero. Each
of these formats gives rise to a definition. These definitions are the topic of the next
section.

Row-Echelon and Reduced Row-Echelon Forms

The first non-zero entry in a row of a matrix (when read from left to right) is called
the leading entry. When the leading entry is 1, we refer to it as a leading
1.

Row-Echelon Form A matrix is said to be in row-echelon form if:

(a)

All entries below each leading entry are 0.

(b)

Each leading entry is in a column to the right of the leading entries in the
rows above it.

(c)

All rows of zeros, if there are any, are located below non-zero rows.

The term row-echelon form can be applied to matrices whether or not they are
augmented matrices (matrices with the vertical bar). For example, both the
coefficient matrix and the augmented matrix in (eq:sys20rowechelon) are in row-echelon form.
Note that the leading entries form a staircase pattern. All entries below the
leading entries are zero, but the entries above the leading entries are not all
zero.

Below are two more examples of matrices in row-echelon form. The leading entries of
each matrix are boxed.

The difference between the coefficient matrix in (eq:sys20rowechelon) and the coefficient matrix in (eq:sys20reducedrowechelon) is
that the leading entries of the matrix in (eq:sys20reducedrowechelon) are all 1’s, and the matrix has zeros above
each leading 1. This motivates our next definition.

Reduced Row-Echelon Form A matrix that is already in row-echelon form is said to
be in reduced row-echelon form if:

(a)

Each leading entry is

(b)

All entries above and below each leading are

The following two matrices are in reduced row-echelon form. Note that there are ’ s
below and above each leading .

When solving linear systems using elementary row operations and the augmented
matrix notation, our goal will be to transform the initial coefficient matrix into its
row-echelon or reduced row-echelon form. The row-echelon form of and the reduced
row-echelon form of are denoted by
respectively.

Solve the system of equations or determine that the system is inconsistent.

We begin by rewriting the system in the augmented matrix form.
Our goal is to carry this matrix to its reduced row-echelon form by means of
elementary row operations. To do this, we will proceed from left to right and use
leading entries to wipe out all entries above and below them.

Our final matrix may not be quite as nice as the one in (eq:sys20reducedrowechelon), but it is in reduced
row-echelon form. Our next step is to convert our augmented matrix back to a
system of equations. We have:
We will rewrite the system as follows:
Now we see that we can assign any value to , then compute , and to obtain a
solution to the system. For example, let , then , and , so is a solution. If we let ,
then , and , so is also a solution. To capture all possibilities, we will let , where is
an arbitrary parameter.

We can think of the solution set in two different ways. First, the solution set is the
set of all points of the form

We can also think of the solution set in geometric terms by observing that

is a set of parametric equations that describes a line in . (See Formula form:paramlinend) This means
that the three hyperplanes given by the equations in the system intersect in a line,
producing infinitely many solutions to the system.

Observe that in Example ex:freevar1, variables , and correspond to the leading in the reduced
row-echelon. We say that , and are the leading variables. Variable is not a
leading variable; we refer to it as a free variable and assign a parameter to
it.

Solve the system of equations or determine that the system is inconsistent.

We rewrite the system in augmented matrix form and transform it to reduced
row-echelon form. We leave the details of the elementary row operations to the reader
and state the final result.

Converting back to a system of linear equations, we get

The last equation in this system clearly has no solutions. We conclude that this
system (and the original system) is inconsistent.

Note that the last row of the reduced row-echelon form in (eq:sys20nosolutionsys) looks like
this
This row corresponds to the equation
which clearly has no solutions.

In general, if the reduced row-echelon form of the augmented matrix contains a
row
we can conclude that the system is inconsistent.

Solve the system of equations or determine that the system is inconsistent.

We rewrite the system in the augmented matrix form and transform it to reduced
row-echelon form. We leave the details of the elementary row operations to the reader
and state the final result.

Converting the augmented matrix back to a system of equations we get

Unlike the last equation in (eq:sys20nosolutions), the last equation in this system has infinitely many
solutions because all values of , and satisfy it. Since the last equation contributes
nothing, we will remove it and rewrite the system as
Variables and correspond to leading in the reduced row-echelon form. So, and are
the leading variables. Variable is a free variable. We let . Solutions to this system are
points of the form
We can also interpret the solutions as lying on a line in given by parametric
equations

This line is the line of intersection of the three planes.

Practice Problems

Determine whether each augmented matrix shown below is in reduced row-echelon
form.

YesNo

YesNo

YesNo

YesNo

YesNo

Fill in the steps that lead to the reduced row-echelon form in Example
ex:nosolutionssys.

Suppose a system of equations has the following reduced row-echelon form
What can you say about the system?

The system is inconsistentThe system
has infinitely many solutionsThe system has a unique solutionWe
would have to examine the original system to make the final determination