MATH 208 Midterm & Final Exams (Latest 2024/ 2025
Updates STUDY BUNDLE PACKAGE WITH SOLUTIONS)
Questions and Verified Answers| 100% Correct| Grade
A- Concordia
Consider the accompanying matrix as the augmented matrix of a linear system. State
in words the next two elementary row operations that should be performed in the
process of solving the system.
1 −6 4 0 −2
0 2 −6 0 5
0 0 1 4 −3
0 0 4 13 −3
What should be the first elementary row operation performed? Select the correct
choice below and, if necessary, fill in the answer box to complete your choice. -
ANSWERReplace row 4 by its sum with −4
times row 3.
What should be the second elementary row operation performed? Select the correct
choice below and, if necessary, fill in the answer box to complete your choice. -
ANSWERScale row 4 by
−1/3.
The augmented matrix of a linear system has been reduced by row operations to the
form shown. Continue the appropriate row operations and describe the solution set
of the original system.
000−3
01−14
0013
172−3
Select the correct choice below and, if necessary, fill in the answer boxes to
complete your choice. - ANSWERThe solution set is empty.
The augmented matrix of a linear system has been reduced by row operations to the
form shown. Continue the appropriate row operations and describe the solution set
of the original system.
1−290
0170
0060
Select the correct choice below and, if necessary, fill in the answer boxes to
complete your choice. - ANSWERThe solution set contains one solution,
(0,0,0).
,The augmented matrix of a linear system has been reduced by row operations to the
form shown. Continue the appropriate row operations and describe the solution set
of the original system.
1−300−4
01−10−6
001−22
00014 - ANSWERThe solution set contains one solution:
(8,4,10,4).
Solve the system.
x1 −6x3 =22
4x1+2x2−9x3=49
x2+5x3=−12 - ANSWERThe unique solution of the system is
(4, 3, −3).
Do the three planes
x1+4x2+x3=4,x2−x3=1,
and 3x1+15x2=8 have at least one common point of intersection? Explain. -
ANSWERThe three planes do not have a common point of intersection.
Determine the value(s) of h such that the matrix is the augmented matrix of a
consistent linear system.
1h4
-5 10 -24 - ANSWERThe matrix is the augmented matrix of a consistent linear system
if h≠−2.
Determine the value(s) of h such that the matrix is the augmented matrix of a
consistent linear system.
1 2 -5
4 h -20 - ANSWERThe matrix is the augmented matrix of a consistent linear system
for every value of h.
Determine the value(s) of h such that the matrix is the augmented matrix of a
consistent linear system.
8 14 h
-4 -7 3 - ANSWERThe matrix is the augmented matrix of a consistent linear system if
h=−6.
Determine whether the statement below is true or false. Justify the answer.
Every elementary row operation is reversible. - ANSWERThe statement is true.
Replacement, interchanging, and scaling are all reversible.
Determine whether the statement below is true or false. Justify the answer.
Elementary row operations on an augmented matrix never change the solution set
of the associated linear system. - ANSWERThe statement is true. Each elementary
row operation replaces a system with an equivalent system.
,Determine whether the statement below is true or false. Justify the answer.
A 5×6 matrix has six rows. - ANSWERThe statement is false. A 5×6 matrix has five
rows and six columns.
Determine whether the statement below is true or false. Justify the answer.
Two matrices are row equivalent if they have the same number of rows. -
ANSWERThe statement is false. Two matrices are row equivalent if there exists a
sequence of elementary row operations that transforms one matrix into the other.
Determine whether the statement below is true or false. Justify the answer.
Two fundamental questions about a linear system involve existence and uniqueness.
- ANSWERThe statement is true. The two fundamental questions are about whether
the solution exists and whether there is only one solution.
Find an equation involving g, h, and k that makes this augmented matrix correspond
to a consistent system.
1 -5 6 g
0 9 -9 h
-3 6 -9 k - ANSWERg+h+k=0
Determine which matrices are in reduced echelon form and which others are only in
echelon form.
a. 1000
0100
0011
b. 01111
00111
00001
00000
c. 0000
1400
0010
0001 - ANSWERa. reduced
b. echelon only
c. neither
Find the general solution of the system whose augmented matrix is given below.
3− 7 4 0
9 −21 12 0
6 −14 8 0 - ANSWERx1=7/3x2−4/3x3
x2 is free
x3 is free
Determine the value(s) of h such that the matrix is the augmented matrix of a
consistent linear system.
1h5
3 6 10 - ANSWERThe matrix is the augmented matrix of a consistent linear system if
, h≠2.
Determine whether the statement below is true or false. Justify the answer.
In some cases, a matrix may be row reduced to more than one matrix in reduced
echelon form, using different sequences of row operations. - ANSWERThe statement
is false. Each matrix is row equivalent to one and only one reduced echelon matrix.
Determine whether the statement below is true or false. Justify the answer.
The echelon form of a matrix is unique. - ANSWERThe statement is false. The
echelon form of a matrix is not unique, but the reduced echelon form is unique.
Determine whether the statement below is true or false. Justify the answer.
A basic variable in a linear system is a variable that corresponds to a pivot column in
the coefficient matrix. - ANSWERThe statement is true. It is the definition of a basic
variable.
Determine whether the statement below is true or false. Justify the answer.
Reducing a matrix to echelon form is called the forward phase of the row reduction
process. - ANSWERThe statement is true. Reducing a matrix to echelon form is called
the forward phase and reducing a matrix to reduced echelon form is called the
backward phase.
Determine whether the statement below is true or false. Justify the answer.
A general solution of a system is an explicit description of all solutions of the system.
- ANSWERThe statement is true. The row reduction algorithm leads directly to an
explicit description of the solution set of a linear system when the algorithm is
applied to the augmented matrix of the system, leading to a general solution of a
system.
Suppose a
3×8 coefficient matrix for a system has three pivot columns. Is the system
consistent? Why or why not? - ANSWERThere is a pivot position in each row of the
coefficient matrix. The augmented matrix will have nine columns and will not have a
row of the form [000000001],
so the system is consistent.
Suppose the coefficient matrix of a system of linear equations has a pivot position in
every row. Explain why the system is consistent. - ANSWERThe system is consistent
because the rightmost column of the augmented matrix is not a pivot column.
Compute each matrix sum or product if it is defined. If an expression is undefined,
explain why. Let
A=3 0 −3
3 −5 3,
B=6 −5 2
2−4−4
C=2 3
Updates STUDY BUNDLE PACKAGE WITH SOLUTIONS)
Questions and Verified Answers| 100% Correct| Grade
A- Concordia
Consider the accompanying matrix as the augmented matrix of a linear system. State
in words the next two elementary row operations that should be performed in the
process of solving the system.
1 −6 4 0 −2
0 2 −6 0 5
0 0 1 4 −3
0 0 4 13 −3
What should be the first elementary row operation performed? Select the correct
choice below and, if necessary, fill in the answer box to complete your choice. -
ANSWERReplace row 4 by its sum with −4
times row 3.
What should be the second elementary row operation performed? Select the correct
choice below and, if necessary, fill in the answer box to complete your choice. -
ANSWERScale row 4 by
−1/3.
The augmented matrix of a linear system has been reduced by row operations to the
form shown. Continue the appropriate row operations and describe the solution set
of the original system.
000−3
01−14
0013
172−3
Select the correct choice below and, if necessary, fill in the answer boxes to
complete your choice. - ANSWERThe solution set is empty.
The augmented matrix of a linear system has been reduced by row operations to the
form shown. Continue the appropriate row operations and describe the solution set
of the original system.
1−290
0170
0060
Select the correct choice below and, if necessary, fill in the answer boxes to
complete your choice. - ANSWERThe solution set contains one solution,
(0,0,0).
,The augmented matrix of a linear system has been reduced by row operations to the
form shown. Continue the appropriate row operations and describe the solution set
of the original system.
1−300−4
01−10−6
001−22
00014 - ANSWERThe solution set contains one solution:
(8,4,10,4).
Solve the system.
x1 −6x3 =22
4x1+2x2−9x3=49
x2+5x3=−12 - ANSWERThe unique solution of the system is
(4, 3, −3).
Do the three planes
x1+4x2+x3=4,x2−x3=1,
and 3x1+15x2=8 have at least one common point of intersection? Explain. -
ANSWERThe three planes do not have a common point of intersection.
Determine the value(s) of h such that the matrix is the augmented matrix of a
consistent linear system.
1h4
-5 10 -24 - ANSWERThe matrix is the augmented matrix of a consistent linear system
if h≠−2.
Determine the value(s) of h such that the matrix is the augmented matrix of a
consistent linear system.
1 2 -5
4 h -20 - ANSWERThe matrix is the augmented matrix of a consistent linear system
for every value of h.
Determine the value(s) of h such that the matrix is the augmented matrix of a
consistent linear system.
8 14 h
-4 -7 3 - ANSWERThe matrix is the augmented matrix of a consistent linear system if
h=−6.
Determine whether the statement below is true or false. Justify the answer.
Every elementary row operation is reversible. - ANSWERThe statement is true.
Replacement, interchanging, and scaling are all reversible.
Determine whether the statement below is true or false. Justify the answer.
Elementary row operations on an augmented matrix never change the solution set
of the associated linear system. - ANSWERThe statement is true. Each elementary
row operation replaces a system with an equivalent system.
,Determine whether the statement below is true or false. Justify the answer.
A 5×6 matrix has six rows. - ANSWERThe statement is false. A 5×6 matrix has five
rows and six columns.
Determine whether the statement below is true or false. Justify the answer.
Two matrices are row equivalent if they have the same number of rows. -
ANSWERThe statement is false. Two matrices are row equivalent if there exists a
sequence of elementary row operations that transforms one matrix into the other.
Determine whether the statement below is true or false. Justify the answer.
Two fundamental questions about a linear system involve existence and uniqueness.
- ANSWERThe statement is true. The two fundamental questions are about whether
the solution exists and whether there is only one solution.
Find an equation involving g, h, and k that makes this augmented matrix correspond
to a consistent system.
1 -5 6 g
0 9 -9 h
-3 6 -9 k - ANSWERg+h+k=0
Determine which matrices are in reduced echelon form and which others are only in
echelon form.
a. 1000
0100
0011
b. 01111
00111
00001
00000
c. 0000
1400
0010
0001 - ANSWERa. reduced
b. echelon only
c. neither
Find the general solution of the system whose augmented matrix is given below.
3− 7 4 0
9 −21 12 0
6 −14 8 0 - ANSWERx1=7/3x2−4/3x3
x2 is free
x3 is free
Determine the value(s) of h such that the matrix is the augmented matrix of a
consistent linear system.
1h5
3 6 10 - ANSWERThe matrix is the augmented matrix of a consistent linear system if
, h≠2.
Determine whether the statement below is true or false. Justify the answer.
In some cases, a matrix may be row reduced to more than one matrix in reduced
echelon form, using different sequences of row operations. - ANSWERThe statement
is false. Each matrix is row equivalent to one and only one reduced echelon matrix.
Determine whether the statement below is true or false. Justify the answer.
The echelon form of a matrix is unique. - ANSWERThe statement is false. The
echelon form of a matrix is not unique, but the reduced echelon form is unique.
Determine whether the statement below is true or false. Justify the answer.
A basic variable in a linear system is a variable that corresponds to a pivot column in
the coefficient matrix. - ANSWERThe statement is true. It is the definition of a basic
variable.
Determine whether the statement below is true or false. Justify the answer.
Reducing a matrix to echelon form is called the forward phase of the row reduction
process. - ANSWERThe statement is true. Reducing a matrix to echelon form is called
the forward phase and reducing a matrix to reduced echelon form is called the
backward phase.
Determine whether the statement below is true or false. Justify the answer.
A general solution of a system is an explicit description of all solutions of the system.
- ANSWERThe statement is true. The row reduction algorithm leads directly to an
explicit description of the solution set of a linear system when the algorithm is
applied to the augmented matrix of the system, leading to a general solution of a
system.
Suppose a
3×8 coefficient matrix for a system has three pivot columns. Is the system
consistent? Why or why not? - ANSWERThere is a pivot position in each row of the
coefficient matrix. The augmented matrix will have nine columns and will not have a
row of the form [000000001],
so the system is consistent.
Suppose the coefficient matrix of a system of linear equations has a pivot position in
every row. Explain why the system is consistent. - ANSWERThe system is consistent
because the rightmost column of the augmented matrix is not a pivot column.
Compute each matrix sum or product if it is defined. If an expression is undefined,
explain why. Let
A=3 0 −3
3 −5 3,
B=6 −5 2
2−4−4
C=2 3
