Skip to main content
\(\newcommand{\bvec}{{\mathbf b}} \newcommand{\cvec}{{\mathbf c}} \newcommand{\dvec}{{\mathbf d}} \newcommand{\evec}{{\mathbf e}} \newcommand{\fvec}{{\mathbf f}} \newcommand{\qvec}{{\mathbf q}} \newcommand{\uvec}{{\mathbf u}} \newcommand{\vvec}{{\mathbf v}} \newcommand{\wvec}{{\mathbf w}} \newcommand{\xvec}{{\mathbf x}} \newcommand{\yvec}{{\mathbf y}} \newcommand{\zvec}{{\mathbf y}} \newcommand{\zerovec}{{\mathbf 0}} \newcommand{\real}{{\mathbb R}} \newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]} \newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]} \newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]} \newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]} \newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]} \newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]} \newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]} \newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]} \newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]} \renewcommand{\span}[1]{\text{Span}\{#1\}} \newcommand{\bcal}{{\cal B}} \newcommand{\ccal}{{\cal C}} \newcommand{\scal}{{\cal S}} \newcommand{\wcal}{{\cal W}} \newcommand{\ecal}{{\cal E}} \newcommand{\coords}[2]{\left\{#1\right\}_{#2}} \newcommand{\gray}[1]{\color{gray}{#1}} \newcommand{\lgray}[1]{\color{lightgray}{#1}} \newcommand{\rank}{\text{rank}} \newcommand{\col}{\text{Col}} \newcommand{\nul}{\text{Nul}} \newcommand{\lt}{<} \newcommand{\gt}{>} \newcommand{\amp}{&} \)

Section1.4Pivots and their influence on solution spaces

By now, we have seen many examples illustrating how the reduced row echelon matrix provides a convenient description of the solution space to a system of linear equations. In this section, we will use this understanding to make some general observations about how certain features of the reduced row echelon matrix reflect the nature of the solution space.

We begin with the following definition.

Definition1.4.1

A pivot position in a matrix \(A\) is the position of a leading entry in the reduced row echelon matrix of \(A\text{.}\)

Preview Activity1.4.1Some basic observations about pivots

  1. Given below is a matrix and its reduced row echelon form. Indicate the pivot positions.

    \begin{equation*} \left[ \begin{array}{rrrr} 2 \amp 4 \amp 6 \amp -1 \\ -3 \amp 1 \amp 5 \amp 0 \\ 1 \amp 3 \amp 5 \amp 1 \\ \end{array} \right] \sim \left[ \begin{array}{rrrr} 1 \amp 0 \amp -1 \amp 0 \\ 0 \amp 1 \amp 2 \amp 0 \\ 0 \amp 0 \amp 0 \amp 1 \\ \end{array} \right] \text{.} \end{equation*}
  2. How many pivot positions can there be in one row? In a \(3\times5\) matrix, what is the largest possible number of pivot positions? Give an example of a matrix that has the largest possible number of pivot positions.

  3. How many pivots can there be in one column? In a \(5\times3\) matrix, what is the largest possible number of pivot positions? Give an example of a matrix that has the largest possible number of pivot positions.

  4. Give an example of a matrix with a pivot position in every row and every column. What is special about such a matrix?

When we have looked at solution spaces of systems of linear equations, we have frequently asked whether there are infinitely many solutions, exactly one solution, or no solutions. We will now break this question down into two separate questions.

Question1.4.2

When we encounter a system of linear equations, we will ask

Existence

Is there a solution to the system of linear equations? If so, we say that the system is consistent; if not, we say it is inconsistent.

Uniqueness

If the system of equations is consistent, is the solution unique or are there infinitely many solutions?

These two questions represent two sides of a coin that appear in many variations throughout our explorations. In this section, we will study how the location of the pivots influence the answers to these two questions. We begin by considering the question concerning the existence of solutions.

Subsection1.4.1The existence of solutions

Activity1.4.2

  1. Shown below are three augmented matrices in reduced row echelon form.

    \begin{equation*} \left[ \begin{array}{rrr|r} 1 \amp 0 \amp 0 \amp 3 \\ 0 \amp 1 \amp 0 \amp 0 \\ 0 \amp 0 \amp 1 \amp -2 \\ 0 \amp 0 \amp 0 \amp 0 \\ \end{array} \right] \end{equation*}

    \begin{equation*} \left[ \begin{array}{rrr|r} 1 \amp 0 \amp 2 \amp 3 \\ 0 \amp 1 \amp -1 \amp 0 \\ 0 \amp 0 \amp 0 \amp 0 \\ 0 \amp 0 \amp 0 \amp 0 \\ \end{array} \right] \end{equation*}

    \begin{equation*} \left[ \begin{array}{rrr|r} 1 \amp 0 \amp 2 \amp 0 \\ 0 \amp 1 \amp -1 \amp 0 \\ 0 \amp 0 \amp 0 \amp 1 \\ 0 \amp 0 \amp 0 \amp 0 \\ \end{array} \right] \end{equation*}

    For each matrix, identify the pivot positions and determine if the corresponding linear system is consistent. Explain how the location of the pivots determine consistency or inconsistency.

  2. Each of these augmented matrices above has a row in which each entry is zero. What, if anything, does the presence of such a row tell us about the consistency of the corresponding linear system?

  3. Give an example of a \(3\times5\) augmented matrix in reduced row echelon form that represents a consistent system. Indicate the pivot positions in your matrix and explain why these pivot positions guarantee a consistent system.

  4. Give an example of a \(3\times5\) augmented matrix in reduced row echelon form that represents an inconsistent system. Indicate the pivot positions in your matrix and explain why these pivot positions guarantee an inconsistent system.

  5. Write the reduced row echelon form of the coefficient matrix of the corresponding linear system in Item d? (Remember that Proposition 1.3.1 says that the reduced row echelon form of the coefficient matrix simply consists of the first four columns of the augmented matrix.) What do you notice about the pivot positions in this coefficient matrix?

  6. Suppose we have a linear system for which the coefficient matrix has the following reduced row echelon form.

    \begin{equation*} \left[ \begin{array}{rrrrr} 1 \amp 0 \amp 0 \amp 0 \amp -1 \\ 0 \amp 1 \amp 0 \amp 0 \amp 2 \\ 0 \amp 0 \amp 1 \amp 0 \amp 0 \\ 0 \amp 0 \amp 0 \amp 1 \amp -3 \\ \end{array} \right] \end{equation*}

    What can you say about the consistency of the linear system?

The third example in Item a above asks us to consider the reduced row echelon matrix

\begin{equation*} \left[ \begin{array}{rrr|r} 1 \amp 0 \amp * \amp 0 \\ 0 \amp 1 \amp * \amp 0 \\ 0 \amp 0 \amp 0 \amp 1 \\ 0 \amp 0 \amp 0 \amp 0 \\ \end{array} \right] \text{.} \end{equation*}

In terms of the variables \(x\text{,}\) \(y\text{,}\) and \(z\text{,}\) the final equation says

\begin{equation*} 0x + 0y + 0z = 0 \text{.} \end{equation*}

If we evaluate the left-hand side with any values of \(x\text{,}\) \(y\text{,}\) and \(z\text{,}\) we get 0, which means that the equation always holds. Therefore, its presence has no effect on the solution space defined by the other three equations.

The third equation, however, says that

\begin{equation*} 0x + 0y + 0z = 1 \text{.} \end{equation*}

Again, if we evaluate the left-hand side with any values of \(x\text{,}\) \(y\text{,}\) and \(z\text{,}\) we get 0 so this equation cannot be satisfied for any \((x,y,z)\text{.}\) This means that the entire system of equations has no solution and is therefore inconsistent.

An equation like this appears in the reduced row echelon matrix as

\begin{equation*} \left[ \begin{array}{rrrr|r} \vdots \amp \vdots \amp \vdots \amp \vdots \amp \vdots \\ 0 \amp 0 \amp \ldots \amp 0 \amp 1 \\ \vdots \amp \vdots \amp \vdots \amp \vdots \amp \vdots \\ \end{array} \right] \text{.} \end{equation*}

The pivot positions make this condition clear: the system is inconsistent if there is a pivot in the rightmost column of the corresponding augmented matrix.

In fact, we will soon see that the system is consistent if there is not a pivot in the rightmost column of the corresponding augmented matrix. This leaves us with the following

This also says something about the pivot positions of the coefficient matrix. Consider an example of an inconsistent system corresponding to the reduced row echelon form of the following augmented matrix

\begin{equation*} \left[ \begin{array}{ccc|c} 1 \amp 0 \amp * \amp 0 \\ 0 \amp 1 \amp * \amp 0 \\ 0 \amp 0 \amp 0 \amp 1 \\ \end{array} \right] \text{.} \end{equation*}

Proposition 1.3.1 says that that the reduced row echelon form of the coefficient matrix is

\begin{equation*} \left[ \begin{array}{ccc} 1 \amp 0 \amp * \\ 0 \amp 1 \amp * \\ 0 \amp 0 \amp 0 \\ \end{array} \right] \text{.} \end{equation*}

This situation can only happen if the coefficient matrix has a row without a pivot position. To turn this around, we see: if every row of the coefficient matrix has a pivot position, then the system must be consistent. For instance, if our system of equations has a coefficient matrix whose reduced row echelon form is

\begin{equation*} \left[ \begin{array}{ccc} 1 \amp 0 \amp 0 \\ 0 \amp 1 \amp 0 \\ 0 \amp 0 \amp 1 \\ \end{array} \right] \text{,} \end{equation*}

then we can guarantee that the system of equations is consistent because there is no way to obtain a pivot in the rightmost column of the augmented matrix.

Subsection1.4.2The uniqueness of solutions

Now that we have studied the role that pivot positions play in the existence of solutions, let's turn to the question of uniqueness.

Activity1.4.3

  1. Here are the three augmented matrices in reduced row echelon form that we considered in the previous section.

    \begin{equation*} \left[ \begin{array}{rrr|r} 1 \amp 0 \amp 0 \amp 3 \\ 0 \amp 1 \amp 0 \amp 0 \\ 0 \amp 0 \amp 1 \amp -2 \\ 0 \amp 0 \amp 0 \amp 0 \\ \end{array} \right] \end{equation*}

    \begin{equation*} \left[ \begin{array}{rrr|r} 1 \amp 0 \amp 2 \amp 3 \\ 0 \amp 1 \amp -1 \amp 0 \\ 0 \amp 0 \amp 0 \amp 0 \\ 0 \amp 0 \amp 0 \amp 0 \\ \end{array} \right] \end{equation*}

    \begin{equation*} \left[ \begin{array}{rrr|r} 1 \amp 0 \amp 2 \amp 0 \\ 0 \amp 1 \amp -1 \amp 0 \\ 0 \amp 0 \amp 0 \amp 1 \\ 0 \amp 0 \amp 0 \amp 0 \\ \end{array} \right] \end{equation*}

    For each matrix, identify the pivot positions and state whether the corresponding system of linear equations is consistent. If the system is consistent, explain whether the solution is unique or whether there are infinitely many solutions.

  2. If possible, give an example of a \(3\times5\) augmented matrix that corresponds to a system of linear equations having a unique solution. If it is not possible, explain why.

  3. If possible, give an example of a \(5\times3\) augmented matrix that corresponds to a system of linear equations having a unique solution. If it is not possible, explain why.

  4. What condition on the pivot positions guarantees that a system of linear equations has a unique solution?

  5. If a system of linear equations has a unique solution, what can we say about the relationship between the number of equations and the number of unknowns?

Let's consider what we've learned in this activity. Since we are interested in the question of whether a consistent linear system has a unique solution or infinitely many, we will only consider consistent systems. By the results of the previous section, this means that there is not a pivot in the rightmost column of the augmented matrix. Here are two possible examples:

\begin{equation*} \left[ \begin{array}{rrr|r} 1 \amp 0 \amp 0 \amp 3 \\ 0 \amp 1 \amp 0 \amp 0 \\ 0 \amp 0 \amp 1 \amp -2 \\ 0 \amp 0 \amp 0 \amp 0 \\ \end{array} \right] \end{equation*}

\begin{equation*} \left[ \begin{array}{rrr|r} 1 \amp 0 \amp 2 \amp 3 \\ 0 \amp 1 \amp -1 \amp 0 \\ 0 \amp 0 \amp 0 \amp 0 \\ 0 \amp 0 \amp 0 \amp 0 \\ \end{array} \right] \end{equation*}

In the first example, we have the equations

\begin{equation*} \begin{alignedat}{4} x_1 \amp \amp \amp \amp \amp {}={} \amp 3 \\ \amp \amp x_2 \amp \amp \amp {}={} \amp 0 \\ \amp \amp \amp \amp x_3\amp {}={} \amp -2 \\ \end{alignedat} \end{equation*}

demonstrating the fact that there is a unique solution \((x_1,x_2,x_3) = (3,0,-2)\text{.}\)

In the second example, we have the equations

\begin{equation*} \begin{alignedat}{4} x_1 \amp \amp \amp {}+{} \amp 2x_3\amp {}={} \amp 3 \\ \amp \amp x_2 \amp {}-{} \amp x_3\amp {}={} \amp 0 \\ \end{alignedat} \end{equation*}

that we may rewrite as

\begin{equation*} \begin{alignedat}{4} x_1 \amp \amp \amp \amp \amp {}={} \amp 3-2x_3 \\ \amp \amp x_2 \amp \amp \amp {}={} \amp x_3 \\ \end{alignedat} \text{.} \end{equation*}

From here, we see that \(x_1\) and \(x_2\) are basic variables that may be expressed in terms of the free variable \(x_3\text{.}\) In this case, the presence of the free variable leads to infinitely many solutions.

Remember that every column of the coefficient matrix corresponds to a variable in our linear system. In the first example, we see that every column of the coefficient contains a pivot position, which means that every variable is uniquely determined. In the second example, the column of the coefficient matrix corresponding to \(x_3\) does not contain a pivot position, which results in \(x_3\) appearing as a free variable. This illustrates the following principle.

When a linear system has a unique solution, every column of the coefficient matrix has a pivot position. Since every row contains at most one pivot position, there must be at least as many rows as columns in the coefficient matrix. Therefore, the linear system has at least as many equations as unknowns, which is something we intuitively suspected in Section 1.

It is reasonable to ask how we choose the free variables. For instance, if we have a single equation

\begin{equation*} x + 2y = 4 \text{,} \end{equation*}

then we may write

\begin{equation*} x = 4-2y \end{equation*}

or, equivalently,

\begin{equation*} y = 2 - \frac12 x \text{.} \end{equation*}

Clearly, either variable may be considered as a free variable in this case.

We will see in the future that we are more interested in the number of free variables rather than in their choice. For convenience, we will adopt the convention of calling the variables corresponding to columns that contain no pivot position free, which allows us to quickly identify them. In particular, the variables \(x_2\) and \(x_4\) appear as free variables in the following linear system:

\begin{equation*} \left[ \begin{array}{rrrr|r} 1 \amp 0 \amp 0 \amp 2 \amp 3 \\ 0 \amp 0 \amp 1 \amp -1 \amp 0 \\ \end{array} \right] \text{.} \end{equation*}

Subsection1.4.3Summary

We have seen how the location of pivot positions, in both the augmented and coefficient matrices, gives vital information about the existence and uniqueness of solutions to linear systems. More specifically,

  • A linear system is inconsistent exactly when a pivot position appears in the rightmost column of the augmented matrix.

  • If a linear system is consistent, the solution is unique when every column of the coefficient matrix contains a pivot position. There are infinitely solutions when there is a column of the coefficient matrix without a pivot position.

  • If a linear system if consistent, the columns of the coefficient matrix containing pivot positions correspond to basic variables and the columns without pivot positions correspond to free variables.

Subsection1.4.4Exercises

1

For each of the augmented matrices in reduced row echelon form given below, determine whether the corresponding linear system is consistent and, if it is, determine whether the solution is unique. If the system is consistent, identify the free variables and the basic variables and give a description of the solution space in parametric form.

  1. \begin{equation*} \left[ \begin{array}{rrrr|r} 0 \amp 1 \amp 0 \amp 0 \amp 2 \\ 0 \amp 0 \amp 1 \amp 0 \amp 3 \\ 0 \amp 0 \amp 0 \amp 1 \amp -2 \\ \end{array} \right] \text{.} \end{equation*}
  2. \begin{equation*} \left[ \begin{array}{rrrr|r} 1 \amp 0 \amp 0 \amp 0 \amp 0 \\ 0 \amp 1 \amp 0 \amp 0 \amp 0 \\ 0 \amp 0 \amp 1 \amp 1 \amp 0 \\ 0 \amp 0 \amp 0 \amp 0 \amp 0 \\ \end{array} \right] \text{.} \end{equation*}
  3. \begin{equation*} \left[ \begin{array}{rrrr|r} 1 \amp 0 \amp 0 \amp 0 \amp 0 \\ 0 \amp 1 \amp 0 \amp 0 \amp 0 \\ 0 \amp 0 \amp 1 \amp 0 \amp 0 \\ 0 \amp 0 \amp 0 \amp 1 \amp 0 \\ 0 \amp 0 \amp 0 \amp 0 \amp 1 \\ \end{array} \right] \text{.} \end{equation*}
  4. \begin{equation*} \left[ \begin{array}{rrr|r} 1 \amp 0 \amp 0 \amp -3 \\ 0 \amp 1 \amp 0 \amp -1 \\ 0 \amp 0 \amp 1 \amp -2 \\ \end{array} \right] \text{.} \end{equation*}
2

Include an example of an appropriate matrix as you justify your responses to the following questions.

  1. Suppose a linear system having six equations and three unknowns is consistent. Can you guarantee that the solution is unique? Can you guarantee that there are infinitely solutions?

  2. Suppose that a linear system having three equations and six unknowns is consistent. Can you guarantee that the solution is unique? Can you guarantee that there are infinitely solutions?

  3. Suppose that a linear system is consistent and has a unique solution. What can you guarantee about the pivot positions in the augmented matrix?

3

Determine whether the following statements are true or false and provide a justification for your response.

  1. If the coefficient matrix of a system of linear equations has a pivot in the rightmost column, then the system is inconsistent.

  2. If a system of equations has two equations and four unknowns, then it must be consistent.

  3. If a system of equations having four equations and three unknowns is consistent, then the solution is unique.

  4. Suppose that a linear system has four equations and four unknowns and that the coefficient matrix has four pivots. Then the linear system is consistent and has a unique solution.

  5. Suppose that a linear system has five equations and three unknowns and that the coefficient matrix has a pivot in every column. The the linear system is consistent and has a unique solution.

4

The following systems contain either one or two parameters.

  1. For what values of the parameter \(k\) is the following system consistent? For which of those values is the solution unique?

    \begin{equation*} \begin{alignedat}{3} -x_1 \amp {}+{} \amp 2x_2 \amp {}={} \amp 3 \\ 2x_1 \amp {}-{} \amp 4x_2 \amp {}={} \amp k \\ \end{alignedat} \text{.} \end{equation*}
  2. For what values of the parameters \(k\) and l is the following system consistent? For which of those values is the solution unique?

    \begin{equation*} \begin{alignedat}{3} 2x_1 \amp {}+{} \amp 4x_2 \amp {}={} \amp 3 \\ -x_1 \amp {}+{} \amp kx_2 \amp {}={} \amp l \\ \end{alignedat} \text{.} \end{equation*}
5

Consider the system of equations described by the following augmented matrix.

\begin{equation*} \left[ \begin{array}{ccc|c} 1 \amp 2 \amp 3 \amp 1 \\ 4 \amp 5 \amp 6 \amp 4 \\ a \amp b \amp c \amp 9 \\ \end{array} \right] \text{.} \end{equation*}
  1. Find a choice for the parameters \(a\text{,}\) \(b\text{,}\) and \(c\) that causes the linear system to be inconsistent. Explain why your choice has this property.

  2. Find a choice for the parameters \(a\text{,}\) \(b\text{,}\) and \(c\) that causes the linear system to have a unique solution. Explain why your choice has this property.

  3. Find a choice for the parameters \(a\text{,}\) \(b\text{,}\) and \(c\) that causes the linear system to have infinitely many solutions. Explain why your choice has this property.

6

A system of equations where the right hand side of every equation is 0 is called homogeneous. The augmented matrix of a homogeneous system, for instance, has the following form:

\begin{equation*} \left[ \begin{array}{cccc|c} * \amp * \amp * \amp * \amp 0 \\ * \amp * \amp * \amp * \amp 0 \\ * \amp * \amp * \amp * \amp 0 \\ \end{array} \right] \text{.} \end{equation*}
  1. Using the concepts we've seen in this section, explain why a homogeneous system of equations must be consistent.

  2. What values for the unknowns are guaranteed to give a solution? Use this to offer another explanation for why a homogeneous system of equations is consistent.

  3. Suppose that a homogeneous system of equations has a unique solution.

    1. Give an example of such a system by writing its augmented matrix in reduced row echelon form.

    2. Write just the coefficient matrix for the example you gave in the previous part. What can you say about the pivot positions in the coefficient matrix? Explain why your observation must hold for any homogeneous system having a unique solution.

    3. If a homogenous system of equations has a unique solution, what can you say about the number of equations compared to the number of unknowns?

7

In a previous math class, you have probably seen the fact that, if we are given two points in the plane, then there is a unique line passing through both of them. In this problem, we will begin with the four points on the left below and ask to find a polynomial that passes through these four points as shown on the right.

<<SVG image is unavailable, or your browser cannot render it>>

<<SVG image is unavailable, or your browser cannot render it>>

A degree three polynomial can be written as

\begin{equation*} p(x) = a + bx + cx^2 + dx^3 \end{equation*}

where \(a\text{,}\) \(b\text{,}\) \(c\text{,}\) and \(d\) are coefficients that we would like to determine. Since we want the polynomial to pass through the point \((3,1)\text{,}\) we should require that

\begin{equation*} p(3) = a + 3b + 9c + 27d = 1 \text{.} \end{equation*}

In this way, we obtain a linear equation for the coefficients \(a\text{,}\) \(b\text{,}\) \(c\text{,}\) and \(d\text{.}\)

  1. Write the four linear equations for the coefficients obtained by requiring that the graph of the polynomial \(p(x)\) passes through the four points above.

  2. Write the augmented matrix corresponding to this system of equations and use the Sage cell below to solve for the coefficients.

  3. Write the polynomial \(p(x)\) that you found and check your work by graphing it in the Sage cell below and verifying that it passes through the four points. To plot a function over a range, you may use a command like plot(1 + x- 2x^2, xmin = -1, xmax = 4).

  4. Rather than looking for a degree three polynomial, suppose we wanted to find a polynomial that passes through the four points and that has degree two, such as

    \begin{equation*} p(x) = a + bx + cx^2 \text{.} \end{equation*}

    Solve the system of equations for the coefficients. What can you say about the existence and uniqueness of the solutions?

  5. Rather than looking for a degree three polynomial, suppose we wanted to find a polynomial that passes through the four points and that has degree four, such as

    \begin{equation*} p(x) = a + bx + cx^2 + dx^3 + ex^4 \text{.} \end{equation*}

    Solve the system of equations for the coefficients. What can you say about the existence and uniqueness of the solutions?

  6. Suppose you had 10 points and you wanted to find a polynomial passing through each of them. What should the degree of the polynomial be to guarantee that there is exactly one such polynomial? Explain your response.