Math 225 Midterm One with
complete verified solutions
2025-2026
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.
If 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 5x4 = 0, which does not by
itself make the system inconsistent.
Suppose the coefficient matrix of a system of linear equations has a pivot
position in every row. Explain why the system is consistent. - answer The
system is consistent because the rightmost column of the augmented
matrix is not a pivot column.
Suppose a 6x8 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 the
system 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.
Choose the correct answer below. - 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
complete verified solutions
2025-2026
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.
If 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 5x4 = 0, which does not by
itself make the system inconsistent.
Suppose the coefficient matrix of a system of linear equations has a pivot
position in every row. Explain why the system is consistent. - answer The
system is consistent because the rightmost column of the augmented
matrix is not a pivot column.
Suppose a 6x8 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 the
system 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.
Choose the correct answer below. - 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
