Theorem. If $$A$$ is an $$m \times n$$ matrix, $$B$$ is an $$n \times m$$ matrix, and $$n < m$$, then $$AB$$ is not invertible.
Proof. If $$AB$$ is invertible, then the homogenous system $$ABX = 0$$ has only the trivial solution $$X=0$$. By Theorem 6 [1], $$B$$ must have a non-trivial solution since it has less rows than columns. Since $$AB$$ is a product of $$B$$ with $$B$$ on the right side, the rows of $$AB$$ are linear combinations of the rows of $$B$$, and so it must have the same solutions as $$B$$ (and therefore a non-trivial one). Thus, $$AB$$ does not have $$X=0$$ as its only solution, and so it is not invertible.
[1] Theorem 6 is the following: If $$A$$ is an $$m \times n$$ matrix with $$m < n$$, then the homogenous system $$AX = 0$$ has a non-trivial solution.