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TBIL Activities for Understanding Linear Algebra

Jennifer Nordstrom

Section 1.4 Pivots and their influence on solution spaces

Recall that a matrix is in reduced row echelon form (RREF) if
  1. The leading term (first nonzero term) of each nonzero row is a 1. Call these terms pivots.
  2. Each pivot is to the right of every higher pivot.
  3. Each term above or below a pivot is zero.
  4. All rows of zeroes are at the bottom of the matrix.

Activity 1.4.0.1.

For each matrix, circle the leading terms, and label it as RREF or not RREF. For the ones not in RREF, find their RREF.
\begin{equation*} A=\left[\begin{array}{ccc|c} 1 & 0 & 0 & 3 \\ 0 & 0 & 1 & -1 \\ 0 & 0 & 0 & 0 \end{array}\right] \end{equation*}
\begin{equation*} B=\left[\begin{array}{ccc|c} 1 & 2 & 4 & 3 \\ 0 & 0 & 1 & -1 \\ 0 & 0 & 0 & 0 \end{array}\right] \end{equation*}
\begin{equation*} C=\left[\begin{array}{ccc|c} 0 & 0 & 0 & 0 \\ 1 & 2 & 0 & 3 \\ 0 & 0 & 1 & -1 \end{array}\right] \end{equation*}

Activity 1.4.0.2.

For each matrix, circle the leading terms, and label it as RREF or not RREF. For the ones not in RREF, find their RREF.
\begin{equation*} D=\left[\begin{array}{ccc|c} 1 & 0 & 2 & -3 \\ 0 & 3 & 3 & -3 \\ 0 & 0 & 0 & 0 \end{array}\right] \end{equation*}
\begin{equation*} E=\left[\begin{array}{ccc|c} 0 & 1 & 0 & 7 \\ 1 & 0 & 0 & 4 \\ 0 & 0 & 0 & 0 \end{array}\right] \end{equation*}
\begin{equation*} F=\left[\begin{array}{ccc|c} 1 & 0 & 0 & 4 \\ 0 & 1 & 0 & 7 \\ 0 & 0 & 1 & 0 \end{array}\right] \end{equation*}

Activity 1.4.0.3.

In this activity we explore possibilities for the number of pivot positions in a matrix.

(a)

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*}

(b)

How many pivot positions can there be in one row?

(c)

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.

(d)

How many pivots can there be in one column?

(e)

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.

(f)

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

Activity 1.4.0.4.

Shown below are three augmented matrices in reduced row echelon form.
\begin{equation*} (i)\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*} (ii)\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*} (iii)\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*}

(a)

For each matrix, identify the pivot positions and determine if the corresponding linear system is (A) consistent or (B) inconsistent. Be able to explain how the location of the pivots determine consistency or inconsistency.

(b)

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?

Activity 1.4.0.5.

(a)

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 be able to explain why these pivot positions guarantee a consistent system.

(b)

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 be able to explain why these pivot positions guarantee an inconsistent system.

(c)

Write the reduced row echelon form of the coefficient matrix of the corresponding linear system in Task 1.4.0.5.b. What do you notice about the pivot positions in this coefficient matrix?

Activity 1.4.0.6.

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 determine about the consistency of the linear system?
  1. It is consistent.
  2. It is inconsistent.
  3. You can't determine anything about consistency.

Activity 1.4.0.7.

Here are the three augmented matrices in reduced row echelon form that we considered previously.
\begin{equation*} (i)\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*} (ii)\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*} (iii)\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 system of linear equations is
  1. consistent with a unique solution.
  2. consistent with infinitely many solutions.
  3. inconsistent with no solutions.

Activity 1.4.0.8.

In this activity, we want to understand the connection between pivot positions and when a system of linear equationa has a unique solution.

(a)

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, be able to explain why.

(b)

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, be able to explain why.

(c)

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

(d)

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?
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