LINEAR ALGEBRA TRUE OR FALSE
EXAM QUESTIONS WITH COMPLETE
ANSWERS
Every elementary row operation is reversible. - Answer-True, because replacement,
interchanging, and scaling are all reversible
A 5x6 matrix has six rows - Answer-False, because a 5x6 matrix has five rows and six
columns
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 substitutes for x1,...,xn, respectively - Answer-False, because the
description applies to a single solution. The solution set consists of all possible solutions
Two fundamental questions about a linear system involve existence and uniqueness -
Answer-True, because two fundamental questions address whether the solution exists
and whether there is only one solution
Two matrices are row equivalent if they have the same number of rows - Answer-False,
because if two matrices are row equivalent it means that there exists a sequence of row
operations that transforms one matrix to the other.
Elementary row operations on an augmented matrix never change the solution set of
the associated linear system - Answer-True, because the elementary row operations
replace a system with an equivalent system.
Two equivalent linear systems can have different solution sets - Answer-False, because
two systems are called equivalent if they have the same solution set.
A consistent system of linear equations has one or more solutions - Answer-True, a
consistent system is defined as a system that has at least one solution.
In some cases, a matrix may be row reduced to more than one matrix in reduced
echelon form, using different sequences of row operations. - 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. -
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. - 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. - 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.
If one row in an echelon form of an augmented matrix is [0 0 0 5 0], then the associated
linear system is inconsistent. - Answer-The statement is false. The indicated row
corresponds to the equation
5x4=0,
which does not by itself make the system inconsistent.
The echelon form of a matrix is unique - Answer-The statement is false. The echelon
form of a matrix is not unique, but the reduced echelon form is unique.
The pivot positions in a matrix depend on whether row interchanges are used in the row
reduction process - Answer-The statement is false. The pivot positions in a matrix are
determined completely by the positions of the leading entries in the nonzero rows of any
echelon form obtained from the matrix.
Reducing a matrix to echelon form is called the forward phase of the row reduction
process. - Answer-The 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.
Whenever a system has free variables, the solution set contains many solutions -
Answer-The statement is false. The existence of at least one solution is not related to
the presence or absence of free variables. If the system is inconsistent, the solution set
is empty.
A general solution of a system is an explicit description of all solutions of the system. -
Answer-The 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.
Another notation for the vector
[-4]
[3]
is [-4 3] - Answer-False. The alternative notation for a (column) vector is
(-4,3),
using parentheses and commas.
The points in the plane corresponding to
EXAM QUESTIONS WITH COMPLETE
ANSWERS
Every elementary row operation is reversible. - Answer-True, because replacement,
interchanging, and scaling are all reversible
A 5x6 matrix has six rows - Answer-False, because a 5x6 matrix has five rows and six
columns
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 substitutes for x1,...,xn, respectively - Answer-False, because the
description applies to a single solution. The solution set consists of all possible solutions
Two fundamental questions about a linear system involve existence and uniqueness -
Answer-True, because two fundamental questions address whether the solution exists
and whether there is only one solution
Two matrices are row equivalent if they have the same number of rows - Answer-False,
because if two matrices are row equivalent it means that there exists a sequence of row
operations that transforms one matrix to the other.
Elementary row operations on an augmented matrix never change the solution set of
the associated linear system - Answer-True, because the elementary row operations
replace a system with an equivalent system.
Two equivalent linear systems can have different solution sets - Answer-False, because
two systems are called equivalent if they have the same solution set.
A consistent system of linear equations has one or more solutions - Answer-True, a
consistent system is defined as a system that has at least one solution.
In some cases, a matrix may be row reduced to more than one matrix in reduced
echelon form, using different sequences of row operations. - 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. -
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. - 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. - 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.
If one row in an echelon form of an augmented matrix is [0 0 0 5 0], then the associated
linear system is inconsistent. - Answer-The statement is false. The indicated row
corresponds to the equation
5x4=0,
which does not by itself make the system inconsistent.
The echelon form of a matrix is unique - Answer-The statement is false. The echelon
form of a matrix is not unique, but the reduced echelon form is unique.
The pivot positions in a matrix depend on whether row interchanges are used in the row
reduction process - Answer-The statement is false. The pivot positions in a matrix are
determined completely by the positions of the leading entries in the nonzero rows of any
echelon form obtained from the matrix.
Reducing a matrix to echelon form is called the forward phase of the row reduction
process. - Answer-The 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.
Whenever a system has free variables, the solution set contains many solutions -
Answer-The statement is false. The existence of at least one solution is not related to
the presence or absence of free variables. If the system is inconsistent, the solution set
is empty.
A general solution of a system is an explicit description of all solutions of the system. -
Answer-The 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.
Another notation for the vector
[-4]
[3]
is [-4 3] - Answer-False. The alternative notation for a (column) vector is
(-4,3),
using parentheses and commas.
The points in the plane corresponding to
