Professor Munkres begins a proof of matrices having inverses only if their number of rows is equal to their number of columns is equal to their rank by
Let $A$ be a $k$ by $n$ matrix. First, suppose $B$ is a right inverse for $A$. Then $A\cdotp B=I_k$. It follows that the system of equations $A\cdotp X = C$ has a solution for arbitrary $C$, for the vector $X=B \cdotp C$ is one such solution. Theorem 6 then implies that $r$ must equal $k$.
Now Theorem 6 states that
If $r<k$, then there exist vectors $C$ in $V_k$ such that the system $A\cdotp X = C$ has no solution.
If $r=k$, then the system $A\cdotp X=C$ always has a solution.
Hence I don't understand why he states that about Theorem 6, given that Theorem 6 states that there may be a solution for $r<k$ as long as the vectors $C$ correspond (They must have some elements = $0$).