Additional Exercises for Ch 2: Systems of Linear Equations
Review Exercises
Graphically, find the point of intersection of the two lines and That is, graph each
line and see where they intersect.
You have a system of equations in two variables, . Explain the geometric
significance of
- (a)
- No solution.
- (b)
- A unique solution.
- (c)
- An infinite number of solutions.
Consider the following augmented matrix in which denotes an arbitrary number and
denotes a nonzero number. Determine whether the given augmented matrix is
consistent. If consistent, is the solution unique?
The solution exists does not exist and is unique not unique .
Consider the following augmented matrix in which denotes an arbitrary number and
denotes a nonzero number. Determine whether the given augmented matrix is
consistent. If consistent, is the solution unique?
The solution exists does not exist
and is [correct]unique not unique
.
Consider the following augmented matrix in which denotes an arbitrary number and
denotes a nonzero number. Determine whether the given augmented matrix is
consistent. If consistent, is the solution unique?
The solution exists does not exist
and is unique not unique
.
Consider the following augmented matrix in which denotes an arbitrary number and
denotes a nonzero number. Determine whether the given augmented matrix is
consistent. If consistent, is the solution unique?
Click the arrow to see the answer.
The third equation implies that . The fourth equation implies that . We conclude that the system is inconsistent.
Suppose a system of equations has fewer equations than variables. Will such a
system necessarily be consistent? If so, explain why and if not, give an example which
is not consistent.
Click the arrow to see the answer.
No. Consider and
If a system of equations has more equations than variables, can it have a solution? If
so, give an example and if not, tell why not.
Click the arrow to see the answer.
These can have a solution. For example, even has an infinite set of solutions.
Find such that
is the augmented matrix of a consistent system.
Click the arrow to see the answer.
Any will work.
Find such that
is the augmented matrix of a consistent system.
Click the arrow to see the answer.
Any will work.
Choose and such that the augmented matrix shown has each of the following:
- (a)
- one solution
- (b)
- no solution
- (c)
- infinitely many solutions
If there will be a unique solution for any . If and there are no solutions. If and then there are infinitely many solutions.
Choose and such that the augmented matrix shown has each of the following:
- (a)
- one solution
- (b)
- no solution
- (c)
- infinitely many solutions
If then there is exactly one solution. If and then there are no solutions. If and then there are infinitely many solutions.
Determine if the system is consistent. If so, is the solution unique?
Click the arrow to see the answer.
There is no solution. The system is inconsistent. You can see this from the augmented matrix.
Determine if the system is consistent. If so, is the solution unique?
Click the arrow to see the answer.
Solution is:
Row reduce the following matrix to row echelon form. Then continue to obtain the
reduced row echelon form.
Click the arrow to see the answer.
Row reduce the following matrix to row echelon form. Then continue to obtain the
reduced row echelon form.
Click the arrow to see the answer.
Row reduce the following matrix to row echelon form. Then continue to obtain the
reduced row echelon form.
Click the arrow to see the answer.
Row reduce the following matrix to row echelon form. Then continue to obtain the
reduced row echelon form.
Click the arrow to see the answer.
Row reduce the following matrix to row echelon form. Then continue to obtain the
reduced row echelon form.
Click the arrow to see the answer.
Row reduce the following matrix to row echelon form. Then continue to obtain the
reduced row echelon form.
Click the arrow to see the answer.
Row reduce the following matrix to row echelon form. Then continue to obtain the
reduced row echelon form.
Click the arrow to see the answer.
Find the solution of the system whose augmented matrix is
Click the arrow to see the answer.
The reduced row echelon form is Therefore, the solution is of the form where .
Find the solution of the system whose augmented matrix is
Click the arrow to see the answer.
The reduced row echelon form is and so the solution is
Find the solution of the system whose augmented matrix is
Click the arrow to see the answer.
The reduced row echelon form is and so .
Find the solution of the system whose augmented matrix is
Click the arrow to see the answer.
The reduced row echelon form is . Therefore, let Then the other variables are given by
Suppose a system of equations has fewer equations than variables and you have
found a solution to this system of equations. Is it possible that your solution is the
only one? Explain.
Suppose a system of linear equations has a augmented matrix and the last
column is a pivot column. Could the system of linear equations be consistent?
Explain.
Click the arrow to see the answer.
No. If the last column is a pivot column, then the last row looks like this: . This would lead to
Suppose the coefficient matrix of a system of equations with variables has
the property that every column is a pivot column. Does it follow that the
system of equations must have a solution? If so, must the solution be unique?
Explain.
The solution exists does not exist and is unique not unique .
Suppose there is a unique solution to a system of linear equations. What must be
true of the pivot columns in the augmented matrix?
Click the arrow to see the answer.
The last column must not be a pivot column. The remaining columns must each be pivot columns.
The steady state temperature, , of a plate solves Laplace’s equation, One way to
approximate the solution is to divide the plate into a square mesh and require the
temperature at each node to equal the average of the temperature at the four
adjacent nodes. In the following picture, the numbers represent the observed
temperature at the indicated nodes. Find the temperature at the interior nodes,
indicated by and . One of the equations is .
Suppose is an matrix. Explain why the rank of is always no larger than
Click the arrow to see the answer.
It is because you cannot have more leading 1’s than columns and you cannot have more leading 1’s than rows.
State whether each of the following scenarios is possible for the system of equations .
If possible, describe the solution set. That is, tell whether there exists a unique
solution, no solution or infinitely many solutions. Here, denotes the augmented
matrix.
- (a)
- is a matrix, and
Click the arrow to see the answer.
Infinite solution set.
- (b)
- is a matrix, and
Click the arrow to see the answer.
This surely can’t happen. If you add in another column, the rank does not get smaller.
- (c)
- is a matrix, and
Click the arrow to see the answer.
You can’t have the rank equal 4 if you only have two columns.
- (d)
- is a matrix, and
Click the arrow to see the answer.
This is possible, but contains a row , making the system inconsistent.
- (e)
- is a matrix, and .
Click the arrow to see the answer.
In this case, there is a unique solution.
Challenge Exercises
We show in Planes in that the graph of an equation is a plane in space when not all
of , , and are zero.
- (a)
- By examining the possible positions of planes in space, show that three equations in three variables can have zero, one, or infinitely many solutions.
- (b)
- Can two equations in three variables have a unique solution? Give reasons for your answer.
Find all solutions to the following systems of linear equations.
- (a)
-
Click the arrow to see the answer.
, , ,
- (b)
-
In each case find (if possible) conditions on , , and such that the system has zero,
one, or infinitely many solutions.
- (a)
-
Click the arrow to see the answer.
If , no solution. If , , , . If and , the unique solution is , ,
- (b)
Show that any two rows of a matrix can be interchanged by using the other two
elementary row operations. (This shows that one of the elementary row operations is
“redundant”.)
Click the arrow to see the answer.
Show that the real system
has a complex solution: , , where . Explain. What happens when such a real system
has a unique solution?
Click the arrow to see the answer.
The (real) solution is , , where is a parameter. The given complex solution occurs when is complex. If the real system has a unique solution, that solution is real because the coefficients and constants are all real.
A man is ordered by his doctor to take units of vitamin A, units of vitamin B, and
units of vitamin C each day. Three brands of vitamin pills are available, and the
number of units of each vitamin per pill are shown in the accompanying
table.
- (a)
- Find all combinations of pills that provide exactly the required amount of vitamins (no partial pills allowed).
- (b)
- If brands 1, 2, and 3 cost 3 cents, 2 cents, and 5 cents per pill, respectively,
find the least expensive treatment.
of brand 1, of brand 2, of brand 3
A restaurant owner plans to use tables seating , tables seating , and tables seating
, for a total of tables. When fully occupied, the tables seat customers. If
only half of the tables, half of the tables, and one-fourth of the tables
are used, each fully occupied, then customers will be seated. Find , , and
.
- (a)
- Show that a matrix with two rows and two columns that is in reduced
row-echelon form must have one of the following forms:
Click the arrow to see the answer.
The leading in the first row must be in column 1 or 2 or not exist.
- (b)
- List the seven reduced row-echelon forms for matrices with two rows and three columns.
- (c)
- List the four reduced row-echelon forms for matrices with three rows and two columns.
An amusement park charges $ for adults, $ for youths, and $ for children. If
people enter and pay a total of $, find the numbers of adults, youths, and
children.
Octave Exercises
Use Octave to check your work on Problems prb:2.18 to prb:2.38. The command is an
easy way to obtain the reduced row echelon form of a matrix. It is more
challenging to go through the steps of Gaussian elimination, but it can be
done.
To use Octave, go to the Sage Math Cell Webpage, copy the code below into the cell, select OCTAVE as the language, and press EVALUATE. The code does the first two steps of prb:2.18. See if you can add one more command which will put the matrix into row echelon form.
% Solving systems using Gaussian elimination
A=[2 -1 3 -1; 1 0 2 1; 1 -1 1 -2];
R=rref(A)
A
A(2,:)=-1/2*A(1,:)+A(2,:)
A(3,:)=-1/2*A(1,:)+A(3,:)
Use Octave to check your work on Problems prb:2.44 to prb:2.54. The command is an easy way to
obtain the rank of a matrix. Of course, you can also use and you obtain even more
information! Problem prb:2.44 is entered in the code below.
To use Octave, go to the Sage Math Cell Webpage, copy the code below into the cell, select OCTAVE as the language, and press EVALUATE.
% Computing the rank of a matrix
A=[4 -16 -1 -5; 1 -4 0 -1; 1 -4 -1 -2]
r=rank(A)
R=rref(A)
Bibliography
The Review Exercises come from the end of Chapter 1 of Ken Kuttler’s A First Course in Linear Algebra. (CC-BY)
Ken Kuttler, A First Course in Linear Algebra, Lyryx 2017, Open Edition, pp. 42–49.
The Challenge Exercises come from the end of Chapter 1 of Keith Nicholson’s Linear Algebra with Applications. (CC-BY-NC-SA)
W. Keith Nicholson, Linear Algebra with Applications, Lyryx 2018, Open Edition, pp. 33–34.