LINEAR ALGEBRA EXAM STUDY
REVIEW QUESTIONS WITH COMPLETE
SOLUTIONS
Is the statement "Every elementary row operation is reversible" true or false? Explain. -
Answer-True, because replacement, interchanging, and scaling are all reversible.
Is the statement "A 5×6 matrix has six rows" true or false? Explain. - Answer-False,
because a 5times×6 matrix has five rows and six columns.
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.
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
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.
REVIEW QUESTIONS WITH COMPLETE
SOLUTIONS
Is the statement "Every elementary row operation is reversible" true or false? Explain. -
Answer-True, because replacement, interchanging, and scaling are all reversible.
Is the statement "A 5×6 matrix has six rows" true or false? Explain. - Answer-False,
because a 5times×6 matrix has five rows and six columns.
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.
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
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.
