Let $D$ be a $n\times n$ real matrix, $n\ge 2$. Which of the following is valid?
$\det(D)=0\Rightarrow \mathrm{rank}(D)=0$
$\det(D)=1\Rightarrow \mathrm{rank}(D)\neq 1$
$\det(D)=1\Rightarrow \mathrm{rank}(D)\neq0$
$\det(D)=n\Rightarrow \mathrm{rank}(D)\neq 1$
Well, (1) is wrong because there is a $3\times 3$ matrix with rank $2$ and determinant $0$, namely $\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{pmatrix}. $
I am confused about the other three: please help!