Expression for a matrix in terms of the other two matrices given three equations

Question

I have two $3 \times 6$ matrices : $\mathbf{A}$ and $\mathbf{B}$ and two $3 \times 1$ vectors $\mathbf{p}$ and $\mathbf{q}$ such that : $\mathbf{A}^T$$\mathbf{p} = \mathbf{0}$ and $\mathbf{B}^T$$\mathbf{q} = \mathbf{0}$. Furthermore, $\mathbf{p} + \mathbf{q} = \mathbf{r}$. The problem was to find a matrix $\mathbf{J}$ as a function of $\mathbf{A}$ and $\mathbf{B}$ such that $\mathbf{J}^T\mathbf{r}=\mathbf{0}$. I could find out $\mathbf{J}$ as $\mathbf{B}^T(\mathbf{A}\mathbf{A}^T)^{-1}(\mathbf{A}\mathbf{A}^T)$ or as $\mathbf{A}^T(\mathbf{B}\mathbf{B}^T)^{-1}(\mathbf{B}\mathbf{B}^T)$. Now in my case, both $\mathbf{A}\mathbf{A}^T$ and $\mathbf{B}\mathbf{B}^T$ are not invertible. Does this mean $\mathbf{J}$ doesn't exist (Can I have a proof for it)? OR is there any other way to express $\mathbf{J}$ in terms of the other two matrices?

Answer 1

You can prove that the rank of $A$ has the same as rank of $AA^T$. Are you sure that these matrices are not invertible?