LINEAR ALGEBRA EXAM 1 REVIEW
QUESTIONS WITH CORRECT ANSWERS
Every matrix is row equivalent to a unique matrix in echelon form. - Answer-False
Reduced row-echelon form (RREF) is unique for every matrix.
Any system of n linear equations in n variables has at most n solutions. - Answer-False
If a system of linear equations has two different solutions, it must have infinitely many
solutions. - Answer-True
If a system of linear equations has no free variables, then it has a unique solution. -
Answer-False
If an augmented matrix [A b] is transformed into [C d] by elementary row operations,
then the equations Ax=b and Cx=d have exactly the same solution sets. - Answer-True
If a system Ax=b has more than one solution, then so does the system Ax=0. - Answer-
True
If A is an mxn matrix and the equation Ax=b is consistent for some b, then the columns
of A span R^m. - Answer-False
If an augmented matrix [A,b] can be transformed by elementary row operations into
reduced echelon form, then the equation Ax=b is consistent. - Answer-False
If matrices A and B are row equivalent, then they have the same reduced echelon form.
- Answer-True
The equation Ax=0 has the trivial solution if and only if there are no free variables. -
Answer-False
If the equation Ax = 0 has the trivial solution it does not imply that there are no free
variables.
If A is an mxn matrix and the equation Ax=b is consistent for every b in R^m, then A has
m pivot columns. - Answer-True
If an mxn matrix A has a pivot position in every row, then the equation Ax=b has a
unique solution for each b in R^m. - Answer-True
If an nxn matrix A has n pivot positions, then the reduced echelon form of A is the nxn
identity matrix. - Answer-False
If 3x3 matrices A and B each have three pivot positions, then A can be transformed into
B by elementary row operations. - Answer-True
QUESTIONS WITH CORRECT ANSWERS
Every matrix is row equivalent to a unique matrix in echelon form. - Answer-False
Reduced row-echelon form (RREF) is unique for every matrix.
Any system of n linear equations in n variables has at most n solutions. - Answer-False
If a system of linear equations has two different solutions, it must have infinitely many
solutions. - Answer-True
If a system of linear equations has no free variables, then it has a unique solution. -
Answer-False
If an augmented matrix [A b] is transformed into [C d] by elementary row operations,
then the equations Ax=b and Cx=d have exactly the same solution sets. - Answer-True
If a system Ax=b has more than one solution, then so does the system Ax=0. - Answer-
True
If A is an mxn matrix and the equation Ax=b is consistent for some b, then the columns
of A span R^m. - Answer-False
If an augmented matrix [A,b] can be transformed by elementary row operations into
reduced echelon form, then the equation Ax=b is consistent. - Answer-False
If matrices A and B are row equivalent, then they have the same reduced echelon form.
- Answer-True
The equation Ax=0 has the trivial solution if and only if there are no free variables. -
Answer-False
If the equation Ax = 0 has the trivial solution it does not imply that there are no free
variables.
If A is an mxn matrix and the equation Ax=b is consistent for every b in R^m, then A has
m pivot columns. - Answer-True
If an mxn matrix A has a pivot position in every row, then the equation Ax=b has a
unique solution for each b in R^m. - Answer-True
If an nxn matrix A has n pivot positions, then the reduced echelon form of A is the nxn
identity matrix. - Answer-False
If 3x3 matrices A and B each have three pivot positions, then A can be transformed into
B by elementary row operations. - Answer-True
