$S$ is a $3\times 3$ real matrices. Then $S$ contains which matrices Let , $S$ be the set of $3\times 3$ real matrices $A$ with $$AA^T=\left(\begin{matrix}1&0&0\\0&0&0\\0&0&0\end{matrix}\right).$$Then the set $S$ contains : 
(A) a nilpotent matrix.
(B) a matrix of rank $1$.
(C) a matrix of rank $2$.
(D) a non-zero skew-symmetric matrix.
Attempt :
Clearly , $\left(\begin{matrix}1&0&0\\0&0&0\\0&0&0\end{matrix}\right)\in S$. So, (B) is TRUE. 
Again $rank(AA^T)=rank(A)$. As , $rank(AA^T)=1$ , so $S$ does not contain a matrix of rank $2$. So (C) is FALSE.
Again we know the rank of a non-zero skew-symmetric matrix can never be $1$. So , $S$ does not contain a non-zero skew-symmetric matrix. So (D) is FALSE.
But I am unable to understand the option (A) whether it is correct or NOT ? Any hint ?
 A: Let $A=\left(\begin{array}{ccc} a&b&c\\d&e&f\\g&h&i\end{array}\right)$ be some $3\times 3$ matrix. You can multiply out $AA^T$ and you will find that its diagonal entries are $a^2+b^2+c^2, d^2+e^2+f^2,$ and $g^2+h^2+i^2$. Therefore we find that if $A\in S$, then $a^2+b^2+c^2=1$ and $d^2+e^2+f^2=g^2+h^2+i^2=0$, i.e. all of the entries $d$ through $h$ must be zero. It actually follows that these are sufficient conditions for $A\in S$ as well, so this completely characterizes the matrices in $S$.
In particular, $A=\left(\begin{array}{ccc}0&1&0\\0&0&0\\0&0&0\end{array}\right)$ is a nilpotent matrix in $S$, the zero matrix (i.e. the only rank zero matrix) is not in $S$, and there is no non-zero skew-symmetric matrix in $S$.
A: If you denote $\vec\alpha_1$, $\vec\alpha_2$, $\vec\alpha_3$ the rows of the matrix $A$, then the matrix $AA^T$ has entries
$$c_{ij} = \langle \vec\alpha_i,\vec\alpha_j \rangle.$$
In particular, you see that $\langle \vec\alpha_2,\vec\alpha_2 \rangle = \langle \vec\alpha_3,\vec\alpha_3\rangle = 0$, which implies that you have two zero rows; $\vec\alpha_2=\vec\alpha_3=\vec0$. 
Now from the first row you see that $\langle \vec\alpha_1,\vec\alpha_1\rangle=1$, so the first row has to be a vector of length $1$.
The above condition (the first row has length $1$, the other rows are zero) completely describe the set $S$. This makes easier checking whether a matrix fulfilling some additional conditions is in $S$. 
NOTE: This is not very different from Alex G.'s answer. I just wanted to stress the expression $c_{ij}=\langle \vec\alpha_i,\vec\alpha_j \rangle$, since the same is true for matrices of arbitrary size and the connection between the products like this and the inner product might be useful in some situations.
