The set $S$ contains...... Let $S$ be the set of 3$\times$3 real matrix $A$ with 
$$A^TA=
\left(
\begin{array}{}
 1 & 0 & 0\\
0 & 0 & 0\\
0& 0 & 0
\end{array}\right).$$ Then the set $S$ contains 


*

*a nilpotent matrix

*a matrix of rank one

*a matrix of rank two

*a non-zero skew-symmetric matrix
Here $S$ is contains a matrix of rank 1, Take $\left(
\begin{array}{}
 1 & 0 & 0\\
0 & 0 & 0\\
0& 0 & 0
\end{array}\right)$ itself. 
I am confusing with other options, help me. 

Your answer to 2 is correct.  
For 1, consider
$$
\pmatrix{0&0&0\\1&0&0\\0&0&0}
$$
For 3, note that $A$ and $A^TA$ have the same rank for any $A$.
For 4: if $A$ is $3 \times 3$, skew-symmetric, and non-zero, then $A$ must have rank $2$ (why?).
