  The paper presents several results that address a fundamental question in
low-rank matrices recovery: how many measurements are needed to recover low
rank matrices? We begin by investigating the complex matrices case and show
that $4nr-4r^2$ generic measurements are both necessary and sufficient for the
recovery of rank-$r$ matrices in $\C^{n\times n}$ by algebraic tools. Thus, we
confirm a conjecture which is raised by Eldar, Needell and Plan for the complex
case. We next consider the real case and prove that the bound $4nr-4r^2$ is
tight provided $n=2^k+r, k\in \Z_+$. Motivated by Vinzant's work, we construct
$11$ matrices in $\R^{4\times 4}$ by computer random search and prove they
define injective measurements on rank-$1$ matrices in $\R^{4\times 4}$. This
disproves the conjecture raised by Eldar, Needell and Plan for the real case.
Finally, we use the results in this paper to investigate the phase retrieval by
projection and show fewer than $2n-1$ orthogonal projections are possible for
the recovery of $x\in \R^n$ from the norm of them.
