Is $AA^T$ invertible when $A$ is a $3\times 4$ matrix of rank $2$?

Question

Let $$ A =\left( \begin{matrix} 1 & 1 & 2 & 2 \\ 1 & -1 & 2 & -4 \\ 2 & 1 & 4 & 1 \\ \end{matrix}\right) $$ Is $AA^T$ invertible? Justify your answer.

I know that the rank of the matrix is 2 and maybe that has something to do with the answer. But I'm not sure.

Answer 1

Recall that $$ \DeclareMathOperator{rank}{rank}\rank(XY)\leq\min\{\rank(X),\rank(Y)\} $$ whenever the matrix product $XY$ is defined. In our example we have $$ \rank(AA^\top)\leq\min\{\rank(A),\rank(A^\top)\}=\rank(A)=2<3 $$ What can we conclude?

Answer 2

Hint: The rank of a product of matrices is no greater than the least rank of those matrices.