$\Longrightarrow$ Assume $A$ is invertible. not sure how to do then
$\Longleftarrow$ Assume all the singular values of $A$ are non-zero.
There exists two orthogonal matrices $U \in \mathbb{R}^{n \times n}, V \in \mathbb{R}^{m\times m}, \Sigma \in \mathbb{R}^{n \times m} \ s.t. \ \Sigma_{1,1} \geq \Sigma_{2,2} \geq ... \geq 0$ such that $A = U\Sigma V^\top$, and $U$ and $V$ are invertible
Since the diagonal of $\Sigma$ is non-zero, $\Sigma$ is invertible $\Rightarrow$ $A=U\Sigma V^\top$ is invertible
Also if now we assume A is invertible. How to show that $$\sigma_1(A)\sigma_1(A^{−1}) \geq 1$$ where $\sigma_1(A)$ and $\sigma_1(A^{−1})$ represents the largest singular value of $A$ and $A^{-1}$.