If $G$ is NOT full rank, does $A = G'G$ always singular? I know that for matrix with full rank, $G$. Matrix $A = G'G$ is non-singular. My question is, does the converse also hold?   

The rank of $G'G$ is equal to the rank of $G$. (One way to see this is to show that the two matrices have the same nullspace: $Gx=0 \iff G'Gx=0$.)
