MATH 225 2024&2025 Final Exam with
correct answers
Augmented Matrix - correct answer -a coefficient matrix with an extra column containing the constant
terms
Consistent system - correct 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? - correct answer -True,
because replacement, interchanging, and scaling are all reversible.
Is the statement "A
5×6
matrix has six rows" true or false? - correct 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? - correct 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. - correct answer -True, because two fundamental questions address whether the
solution exists and whether there is only one solution.
Echelon Form - correct 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 - correct 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? - correct 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? - correct 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? - correct 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? - correct 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? - correct 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. - correct
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? - correct answer -Yes, because the rightmost column
of the augmented matrix is not a pivot column.
HW 2 #7 - correct answer -Rework
Existence and Uniqueness Theorem - correct 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? - correct 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. - correct 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. - correct answer -False. Span{u,v} includes linear combinations of both u and v.
Any list of five real numbers is a vector in R5 - correct 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} - correct 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.
correct answers
Augmented Matrix - correct answer -a coefficient matrix with an extra column containing the constant
terms
Consistent system - correct 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? - correct answer -True,
because replacement, interchanging, and scaling are all reversible.
Is the statement "A
5×6
matrix has six rows" true or false? - correct 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? - correct 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. - correct answer -True, because two fundamental questions address whether the
solution exists and whether there is only one solution.
Echelon Form - correct 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 - correct 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? - correct 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? - correct 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? - correct 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? - correct 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? - correct 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. - correct
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? - correct answer -Yes, because the rightmost column
of the augmented matrix is not a pivot column.
HW 2 #7 - correct answer -Rework
Existence and Uniqueness Theorem - correct 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? - correct 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. - correct 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. - correct answer -False. Span{u,v} includes linear combinations of both u and v.
Any list of five real numbers is a vector in R5 - correct 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} - correct 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.
