1. Every system of linear equations with 4 equations in 7 variables (unknowns) has infinitely many solutions.
2. Every homogeneous system of linear equations with 4 equations in 7 variables must have a 3-parameter family of solutions.
3. Every homogeneous system of linear equations with 7 equations in 7 unknowns has a unique solution.
4. No system of linear equations with 7 equations in 4 variables has a solution.
5. Every system of linear equations with 7 equations in 7 variables and whose matrix of coefficients is invertible has a unique solution.
Which of these statement(s) are true?

Question

1. Every system of linear equations with 4 equations in 7 variables (unknowns) has infinitely many solutions.
2. Every homogeneous system of linear equations with 4 equations in 7 variables must have a 3-parameter family of solutions.
3. Every homogeneous system of linear equations with 7 equations in 7 unknowns has a unique solution.
4. No system of linear equations with 7 equations in 4 variables has a solution.
5. Every system of linear equations with 7 equations in 7 variables and whose matrix of coefficients is invertible has a unique solution.
Which of these statement(s) are true?

ybarrap · Answer

Hints: 
* If there are free variables, the number of solutions is infinite
 * What are the conditions for invertibility of a matrix, what does it imply about the system of equations it solves? 
* Not every matrix is invertible, even one representing a system with 7 equations and 7 unknowns 
* If there is any chance of a unique solutions, there MUST be 7 equations and 7 unknowns at least. There is also a requirement on the matrix that represents this system -- "invertibility" 

I don't know what a '3-parameter family of solutions' represents. It may be what I call "free variables." If so, then the "every" part of that particular question is what you should focus on. Should the "non-free" variables be independent?