Bounding area of the image of a parallelogram under $M \in GL(n,\mathbb{R})$, $n>2$, using singular values

Question

It is well-known that under a linear transformation $M \in GL(n,\mathbb{R})$, $n$-dimensional parallelepipeds of unit volume are mapped to parallelepipeds with volume $\det M=\alpha_1(M)\ldots \alpha_n(M)$, where $\det$ is the determinant and the $\alpha_i$ are the singular values in non-increasing order. My question is whether, under $M$, the area of the image of a $\textit{parallelogram}$ with unit area (and one vertex at the origin if necessary) is bounded by $\alpha_1(M) \alpha_2(M)$.