How to prove a matrix not invertible?

Question

How can I prove that A*B is not invertible when A (mXn) and B (nXm)

while (m>n)?

Answer 1

HINT: Complete $A$ to $\tilde A$ by adding on the right $m-n$ columns consisting of $0$'s, and $B$ to $\tilde B$ by adding below $m-n$ rows consisting of $0$'s. Observe that $A\cdot B = \tilde A \cdot \tilde B$. Now clearly $\det \tilde A = \det \tilde B = 0$, so $\det AB = 0$.

Answer 2

Hint: A Matrix is invertible iff it is square and has full Rank

Answer 3

Because then $\operatorname{rank}(AB)\leqslant n<m$. If $AB$ was invertible, then its rank would be equal to $m$.

Answer 4

The product matrix $A\cdot B$ is the matrix associated to an endomorphism of $\Bbb R^m$ which is--by construction--a composition $$ \Bbb R^m\longrightarrow\Bbb R^n\longrightarrow\Bbb R^m. $$ Since $n<m$ the second map cannot be surjective, and so neither the composition. Therefore the composition is not an automorphism and the associated matrix cannot be invertible.