  We address the detection of a low rank $n\times n$deterministic matrix
$\mathbf{X}_{0}$ from the noisy observation ${\bf X}_{0}+{\bf Z}$ when
$n\to\infty$, where ${\bf Z}$ is a complex Gaussian random matrix with
independent identically distributed $\mathcal{N}_{c}(0,\frac{1}{n})$ entries.
Thanks to large random matrix theory results, it is now well-known that if the
largest singular value $\lambda_{1}$ of ${\bf X}_{0}$ verifies $\lambda_{1}>1$,
then it is possible to exhibit consistent tests. In this contribution, we prove
a contrario that under the condition $\lambda_{1}<1$, there are no consistent
tests. Our proof is rather simple, inspired by previous works devoted to the
case of rank 1 matrices ${\bf X}_{0}$.
