MATH 308 MIDTERM FREQUENTLY TESTED ACTUAL
EXAM QUESTIONS WITH CORRECT ANSWERS
Question 1.
A linear system with three equations and two variables must be inconsistent.
ANSWER
False
Question 2.
A linear system with three equations and five variables must be consistent.
ANSWER
False.
Question 3.
A system in echelon form can have more variables than equations.
ANSWER
True.
Question 4.
If two matrices are equivalent, then one can be transformed into the other with a sequence of
elementary row operations.
ANSWER
True, by definition of equivalent matrices.
Question 5.
Different sequences of row operations can lead to different echelon forms for the same matrix.
ANSWER
True. For example, 2 4 (1/2)R1 + R2 => R2~ 1 1 2 4 0 1 And 2 4 1 1 R1 Ô R2~ 1 1 2 4 2R1 +
R2 right double arrow implies R2~ 1 1 0 2
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, Question 6.
Different sequences of row operations can lead to different reduced echelon forms for the same
matrix.
ANSWER
False, by Theorem 1.5.
Question 7.
If a linear system has four equations and seven variables, then it must have infinitely many
solutions.
ANSWER
False, It could be inconsistent, and therefore have no solutions, as with the system
x1+x2+x3+x4 +x5+x6+x7=0 x1+x2+x3 =1 x4 +x5 =1 x6+ x7=1
Question 8.
Every linear system with free variables has infinitely many solutions.
ANSWER
True, a free variable can take any value, and so there are infinitely many solutions.
Question 9.
Any linear system with more variables than equations cannot have a unique solution.
ANSWER
True. If it is consistent, there will be at least one free variable, and hence infinitely many
solutions.
Question 10.
If a linear system has the same number of equations and variables, then it must have a unique
solution.
ANSWER
False. For example, the system x1 + x2 = 0 x1 + x2 = 1 has no solutions. And the system x1 +
x2 = 1 2x1 + 2x2 = 2 has infinitely many solutions.
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EXAM QUESTIONS WITH CORRECT ANSWERS
Question 1.
A linear system with three equations and two variables must be inconsistent.
ANSWER
False
Question 2.
A linear system with three equations and five variables must be consistent.
ANSWER
False.
Question 3.
A system in echelon form can have more variables than equations.
ANSWER
True.
Question 4.
If two matrices are equivalent, then one can be transformed into the other with a sequence of
elementary row operations.
ANSWER
True, by definition of equivalent matrices.
Question 5.
Different sequences of row operations can lead to different echelon forms for the same matrix.
ANSWER
True. For example, 2 4 (1/2)R1 + R2 => R2~ 1 1 2 4 0 1 And 2 4 1 1 R1 Ô R2~ 1 1 2 4 2R1 +
R2 right double arrow implies R2~ 1 1 0 2
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, Question 6.
Different sequences of row operations can lead to different reduced echelon forms for the same
matrix.
ANSWER
False, by Theorem 1.5.
Question 7.
If a linear system has four equations and seven variables, then it must have infinitely many
solutions.
ANSWER
False, It could be inconsistent, and therefore have no solutions, as with the system
x1+x2+x3+x4 +x5+x6+x7=0 x1+x2+x3 =1 x4 +x5 =1 x6+ x7=1
Question 8.
Every linear system with free variables has infinitely many solutions.
ANSWER
True, a free variable can take any value, and so there are infinitely many solutions.
Question 9.
Any linear system with more variables than equations cannot have a unique solution.
ANSWER
True. If it is consistent, there will be at least one free variable, and hence infinitely many
solutions.
Question 10.
If a linear system has the same number of equations and variables, then it must have a unique
solution.
ANSWER
False. For example, the system x1 + x2 = 0 x1 + x2 = 1 has no solutions. And the system x1 +
x2 = 1 2x1 + 2x2 = 2 has infinitely many solutions.
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