I have a question and a proposed solution - Please tell me if I'm correct.
Problem: Prove that if $A$ and $B$ are real matrices and the system of equations $AX=B$ has more than one solution, then it has infinitely many.
Solution: Assume that the system of equations $AX=B$ has more than one solution. This means that the reduced row echelon matrix form of the solution to the equation $AX=B$ has at least one free variable, because if all of the variables were pivot variables, then we would be left with a set of unique values for the variables and hence one solution. Therefore, with at least one free variable, we have that the variable(s) can vary over the real numbers. Hence, there are infinitely many solutions.