MATH 225 Final Exam Questions With
Guaranteed Pass Solutions.
Augmented Matrix - Answer a coefficient matrix with an extra column containing the constant terms
Consistent system - Answer a system of equations or inequalities that has at least one solution
Is the statement "Every elementary row operation is reversible" true or false? - Answer True, because
replacement, interchanging, and scaling are all reversible.
Is the statement "A
5×6
matrix has six rows" true or false? - Answer False, because a
5×6
matrix has five rows and six columns.
Is the statement "The solution set of a linear system involving variables
x1...xn is a list of numbers (s1...sn) that makes each equation in the system a true statement when the
values s1....sn are subbed for x1...xn respectively" true or false? - Answer False, because the
description applies to a single solution. The solution set consists of all possible solutions.
Is the statement "Two fundamental questions about a linear system involve existence and uniqueness"
true or false? Explain. - Answer True, because two fundamental questions address whether the
solution exists and whether there is only one solution.
Echelon Form - Answer 1. All nonzero rows are above any rows of all zeros.
2. Each leading entry of a row is in a column to the right of the leading entry of the row above it.
3. All entries in a column below a leading entry are zeros.
,Reduced Echelon Form - Answer Same as echelon form, except all leading entries are 1; each leading 1
is the only non-zero entry in its row; there is only one unique reduced echelon form for every matrix
In some cases, a matrix may be row reduced to more than one matrix in reduced echelon form, using
different sequences of row operations.
Is this statement true or false? - Answer The statement is false. Each matrix is row equivalent to one
and only one reduced echelon matrix.
The row reduction algorithm applies only to augmented matrices for a linear system.
Is this statement true or false? - Answer The statement is false. The algorithm applies to any matrix,
whether or not the matrix is viewed as an augmented matrix for a linear system.
A basic variable in a linear system is a variable that corresponds to a pivot column in the coefficient
matrix.
Is this statement true or false? - Answer The statement is true. It is the definition of a basic variable.
Finding a parametric description of the solution set of a linear system is the same as solving the system.
Is this statement true or false? - Answer The statement is false. The solution set of a linear system can
only be expressed using a parametric description if the system has at least one solution.
f one row in an echelon form of an augmented matrix is [0 0 0 5 0] then the associated linear system is
inconsistent.
Is this statement true or false? - Answer The statement is false. The indicated row corresponds to the
equation
5x 4x4equals=0,
which does not by itself make the system inconsistent.
What must be true of a linear system for it to have a unique solution? Select all that apply. - Answer
The system is consistent, the system has no free variables
Use the augmented matrix to determine if the linear system is consistent. Is the linear system
represented by the augmented matrix consistent? - Answer Yes, because the rightmost column of the
augmented matrix is not a pivot column.
, HW 2 #7 - Answer Rework
Existence and Uniqueness Theorem - Answer A linear system is consistent if and only if the rightmost
column of the augmented matrix is not a pivot column. That is, if and only if an echelon form of the
augmented matrix has no row of the form [0...0 b] with b nonzero.
Suppose a
6*8 coefficient matrix for a system has
six pivot columns. Is the system consistent? Why or why not? - Answer There 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 [0 0 0 0 0 0 0 0 1], so it is consistent
A system of linear equations with fewer equations than unknowns is sometimes called an
underdetermined system. Can such a system have a unique solution? Explain. - Answer No, it cannot
have a unique solution. Because there are more variables than equations, there must be at least one free
variable. If the linear system is consistent and there is at least one free variable, the solution set contains
infinitely many solutions. If the linear system is inconsistent, there is no solution.
a. When u and v are nonzero vectors, Span{u,v} contains only the line through u and the line through v
and the origin. - Answer False. Span{u,v} includes linear combinations of both u and v.
Any list of five real numbers is a vector in R5 - Answer True. R5 denotes the collection of all lists of five
real numbers
Asking whether the linear system corresponding to an augmented matrix [a1 a2 a3 b] had a solution
amounts to asking whether b is in span {a1, a2, a3} - Answer True. An augmented matrix has a solution
when the last column can be written as a linear combination of the other columns. A linear system
augmented has a solution when the last column of its augmented matrix can be written as a linear
combination of the other columns.
The weights c1...cp in a linear combination c1v1+....cpvp cannot all be zero - Answer False. Setting all
the weights equal to zero results in the vector 0.
Guaranteed Pass Solutions.
Augmented Matrix - Answer a coefficient matrix with an extra column containing the constant terms
Consistent system - Answer a system of equations or inequalities that has at least one solution
Is the statement "Every elementary row operation is reversible" true or false? - Answer True, because
replacement, interchanging, and scaling are all reversible.
Is the statement "A
5×6
matrix has six rows" true or false? - Answer False, because a
5×6
matrix has five rows and six columns.
Is the statement "The solution set of a linear system involving variables
x1...xn is a list of numbers (s1...sn) that makes each equation in the system a true statement when the
values s1....sn are subbed for x1...xn respectively" true or false? - Answer False, because the
description applies to a single solution. The solution set consists of all possible solutions.
Is the statement "Two fundamental questions about a linear system involve existence and uniqueness"
true or false? Explain. - Answer True, because two fundamental questions address whether the
solution exists and whether there is only one solution.
Echelon Form - Answer 1. All nonzero rows are above any rows of all zeros.
2. Each leading entry of a row is in a column to the right of the leading entry of the row above it.
3. All entries in a column below a leading entry are zeros.
,Reduced Echelon Form - Answer Same as echelon form, except all leading entries are 1; each leading 1
is the only non-zero entry in its row; there is only one unique reduced echelon form for every matrix
In some cases, a matrix may be row reduced to more than one matrix in reduced echelon form, using
different sequences of row operations.
Is this statement true or false? - Answer The statement is false. Each matrix is row equivalent to one
and only one reduced echelon matrix.
The row reduction algorithm applies only to augmented matrices for a linear system.
Is this statement true or false? - Answer The statement is false. The algorithm applies to any matrix,
whether or not the matrix is viewed as an augmented matrix for a linear system.
A basic variable in a linear system is a variable that corresponds to a pivot column in the coefficient
matrix.
Is this statement true or false? - Answer The statement is true. It is the definition of a basic variable.
Finding a parametric description of the solution set of a linear system is the same as solving the system.
Is this statement true or false? - Answer The statement is false. The solution set of a linear system can
only be expressed using a parametric description if the system has at least one solution.
f one row in an echelon form of an augmented matrix is [0 0 0 5 0] then the associated linear system is
inconsistent.
Is this statement true or false? - Answer The statement is false. The indicated row corresponds to the
equation
5x 4x4equals=0,
which does not by itself make the system inconsistent.
What must be true of a linear system for it to have a unique solution? Select all that apply. - Answer
The system is consistent, the system has no free variables
Use the augmented matrix to determine if the linear system is consistent. Is the linear system
represented by the augmented matrix consistent? - Answer Yes, because the rightmost column of the
augmented matrix is not a pivot column.
, HW 2 #7 - Answer Rework
Existence and Uniqueness Theorem - Answer A linear system is consistent if and only if the rightmost
column of the augmented matrix is not a pivot column. That is, if and only if an echelon form of the
augmented matrix has no row of the form [0...0 b] with b nonzero.
Suppose a
6*8 coefficient matrix for a system has
six pivot columns. Is the system consistent? Why or why not? - Answer There 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 [0 0 0 0 0 0 0 0 1], so it is consistent
A system of linear equations with fewer equations than unknowns is sometimes called an
underdetermined system. Can such a system have a unique solution? Explain. - Answer No, it cannot
have a unique solution. Because there are more variables than equations, there must be at least one free
variable. If the linear system is consistent and there is at least one free variable, the solution set contains
infinitely many solutions. If the linear system is inconsistent, there is no solution.
a. When u and v are nonzero vectors, Span{u,v} contains only the line through u and the line through v
and the origin. - Answer False. Span{u,v} includes linear combinations of both u and v.
Any list of five real numbers is a vector in R5 - Answer True. R5 denotes the collection of all lists of five
real numbers
Asking whether the linear system corresponding to an augmented matrix [a1 a2 a3 b] had a solution
amounts to asking whether b is in span {a1, a2, a3} - Answer True. An augmented matrix has a solution
when the last column can be written as a linear combination of the other columns. A linear system
augmented has a solution when the last column of its augmented matrix can be written as a linear
combination of the other columns.
The weights c1...cp in a linear combination c1v1+....cpvp cannot all be zero - Answer False. Setting all
the weights equal to zero results in the vector 0.
