Pivots and their influence on solution spaces

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

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

2. Each of the augmented matrices above has a row in which each entry is zero. What, if anything, does the presence of such a…

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

   For each matrix, identify the pivot positions and state whether the corresponding linear system 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 corresp…

For each of the augmented matrices in reduced row echelon form given below, determine whether the corresponding linear system is consistent and, if so, 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. .

2. .

3. .

4. .

For each of the following linear systems, determine whether the system is consistent, and, if so, determine whether there are infinitely many solutions.

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 variables is consistent. Can you guarantee that the solution is unique? Can you guarantee that there are infinitely many solutions?

2. Suppose that a linear system having three equations and six variables is consistent. Can you guarantee that the s…

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

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

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

3. If a linear system having four equations and three variables is consistent, then the solution i…

The following linear 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? .

2. For what values of the parameters $k$ and l is the following system consistent? For which of those values is the solution unique? .

Consider the linear system described by the following augmented matrix. .

1. Find a choice for the parameters $a$, $b$, and $c$ that causes the linear system to be inconsistent. Explain why your choice has this property.

2. Find a choice for the parameters $a$, $b$, 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 $a$, $b$, and $c$ t…

A linear system 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: .

1. Using the concepts we've seen in this section, explain why a homogeneous linear system must be consistent.

2. What values for the variables are guaranteed to give a solution? Use this to offer another explanation for why a homogene…