(Key: correct, incorrect, partially correct.)

- What are all the solutions to the system and ?
- can be arbitrary, and is equal to .
- CORRECT.
- and are both zero.
- There are no solutions.
- and .
- INCORRECT. This is only one of the solutions.
- and .
- PARTIALLY. These are valid equations, but this does not quite answer the question.
- and , or and .
- and .
- What are all the solutions to the system and ?
- and .
- CORRECT.
- and .
- There are no solutions.
- There are infinitely many solutions.
- and , or and .
- can be arbitrary, and is equal to .
- can be arbitrary, and is equal to .
- What are all the solutions to the system and ?
- There are no solutions.
- CORRECT.
- There are infinitely many solutions.
- and .
- can be arbitrary, and is equal to .
- can be arbitrary, and is equal to .
- This question cannot be answered correctly.
- .
- INCORRECT. This does not answer the question.
- If one is solving three linear equations involving two unknowns, what happens?
- Usually there will be no solution, but occasionally there will be one or more solutions.
- CORRECT.
- There will never be a solution.
- There will always be a solution.
- There will always be infinitely many solutions.
- Usually there will be one solution, but occasionally there will be no solutions or infinitely many solutions.
- Usually there will be infinitely many solutions, but occasionally there will be one or no solutions.
- Anything can happen.
- There will always be exactly one solution.
- If one is solving two linear equations involving three unknowns, what happens?
- Usually there will be no solution, but occasionally there will be one or more solutions.
- There will never be a solution.
- There will always be a solution.
- There will always be infinitely many solutions.
- Usually there will be one solution, but occasionally there will be no solutions or infinitely many solutions.
- Usually there will be infinitely many solutions, but occasionally there will be one or no solutions.
- Usually there will be infinitely many solutions, but occasionally there will be no solutions.
- CORRECT.
- There will always be exactly one solution.
- If one is solving three linear equations involving three unknowns, what happens?
- Usually there will be no solution, but occasionally there will be one or more solutions.
- There will never be a solution.
- There will always be a solution.
- There will always be infinitely many solutions.
- Usually there will be one solution, but occasionally there will be no solutions or infinitely many solutions.
- CORRECT.
- Usually there will be infinitely many solutions, but occasionally there will be one or no solutions.
- Anything can happen.
- There will always be exactly one solution.
- What is the complete relationship between homogeneous linear systems of equations, and the zero solution (all unknowns equal to zero)?
- The zero solution is always a solution to homogeneous linear systems, and never a solution to inhomogeneous linear systems.
- CORRECT.
- The zero solution is always a solution to both homogeneous and inhomogeneous linear systems.
- The zero solution is always a solution to homogeneous linear systems, but could also be a solution to inhomogeneous linear systems.
- The zero solution is never a solution to inhomogeneous linear systems, and may or may not be a solution to homogeneous linear systems.
- The zero solution can be a solution to both homogeneous and inhomogeneous linear systems, but only if the equations are solvable.
- If a solution to a homogeneous linear system exists at all, it will be the zero solution.
- If a solution to a homogeneous linear system exists at all, then the zero solution will be a solution.
- PARTIALLY. Homogeneous linear systems always have at least one solution, namely the zero solution.
- If one is solving three homogeneous equations involving two unknowns, what happens?
- Usually the zero solution is the only solution, but occasionally one has more solutions.
- CORRECT.
- Usually one has no solutions, but occasionally one has one or infinitely many solutions.
- The zero solution is the only solution.
- There are no solutions.
- One can get different answers, depending on how you approach the problem.
- One has infinitely many solutions, including the zero solution.
- One usually has infinitely many solutions, but occasionally one just has only the zero solution.
- If one is solving two homogeneous equations involving three unknowns, what happens?
- Usually the zero solution is the only solution, but occasionally one has more solutions.
- Usually one has no solutions, but occasionally one has one or infinitely many solutions.
- The zero solution is the only solution.
- There are no solutions.
- One can get different answers, depending on how you approach the problem.
- One has infinitely many solutions, including the zero solution.
- CORRECT.
- One usually has infinitely many solutions, but occasionally one just has only the zero solution.
- If one is solving three homogeneous equations involving three unknowns, what happens?
- Usually the zero solution is the only solution, but occasionally one has more solutions.
- CORRECT.
- Usually one has no solutions, but occasionally one has one or infinitely many solutions.
- The zero solution is the only solution.
- There are no solutions.
- One can get different answers, depending on how you approach the problem.
- One has infinitely many solutions, including the zero solution.
- One usually has infinitely many solutions, but occasionally one just has only the zero solution.
- If a linear system has three equations in four unknowns, then
- The rank of this system can be any number from zero to three.
- CORRECT.
- The rank of this system is three.
- The rank of this system is four.
- The rank of this system can be any number from zero to four.
- The rank of this system can be three or four.
- The rank of this system is twelve.
- The rank of the system is at least three.
- If a linear system has four equations in three unknowns, then
- The rank of this system can be any number from zero to three.
- CORRECT.
- The rank of this system is three.
- The rank of this system is four.
- The rank of this system can be any number from zero to four.
- The rank of this system can be three or four.
- The rank of this system is twelve.
- The rank of the system is at least three.
- If a linear system has four unknowns and has rank three, then
- There are infinitely many solutions, unless the system is inconsistent, in which case there are no solutions.
- CORRECT.
- There are infinitely many solutions.
- There are no solutions (system is inconsistent).
- There is exactly one solution.
- There is either one solution or infinitely many solutions.
- There is either one solution or no solution.
- Anything can happen.
- If a linear system has four unknowns and has rank four, then
- There are infinitely many solutions, unless the system is inconsistent, in which case there are no solutions.
- There are infinitely many solutions.
- There are no solutions (system is inconsistent).
- There is either one solution or no solution.
- CORRECT.
- There is either one solution or infinitely many solutions.
- There is exactly one solution.
- Anything can happen.
- If a linear system has four unknowns, four equations and has rank four, then
- There are infinitely many solutions, unless the system is inconsistent, in which case there are no solutions.
- There are infinitely many solutions.
- There are no solutions (system is inconsistent).
- There is exactly one solution.
- CORRECT.
- There is either one solution or infinitely many solutions.
- There is either one solution or no solution.
- Anything can happen.

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