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?

Let us analyze each statement for a system of linear equations $Ax = b$ with $m$ equations and $n$ variables:

• Statement I is False: If $m < n$ (underdetermined system), the system is not guaranteed to have a solution. For example, the system consisting of $x_1 + x_2 + x_3 = 1$ and $x_1 + x_2 + x_3 = 2$ has $m = 2$ and $n = 3$, but has no solution because the equations are contradictory (inconsistent).

• Statement II is False: If $m > n$ (overdetermined system), a system can still have a solution. For instance, any homogeneous system $Ax = 0$ with $m > n$ always has the trivial solution $x = 0$.

• Statement III is True: If $m = n$, there exists a system that has a solution (in fact, if $\det(A) \neq 0$, it has a unique solution, e.g., $I_n x = b$).

Therefore, only statement III is true, making (C) the correct option.