Consider the systems, each consisting of m linear equations in n variables. I. If m < n, then all such systems have a solution. II. If m > n, then none of these systems has a solution. III. If m = n, then there exists a system which has a solution. Which one of the following is CORRECT? A. I, II and III are true B. Only II and III are true C. Only III is true D. None of them is true

The correct answer is: C. Only III is true.

I. If m < n, then all such systems have a solution.

This is false. If m < n, then the system of equations is said to be underdetermined. In this case, there are infinitely many solutions to the system.

II. If m > n, then none of these systems has a solution.

This is true. If m > n, then the system of equations is said to be overdetermined. In this case, the system of equations is inconsistent and has no solution.

III. If m = n, then there exists a system which has a solution.

This is true. If m = n, then the system of equations is said to be exactly determined. In this case, the system of equations is consistent and has a unique solution.

Therefore, only III is true.