TITLE: Number of possibilities of $10\times10$ matrix
QUESTION [7 upvotes]: If $A$ is a $10\times10$ matrix with entries from the set $\{0, 1, 2, 3\}$ and if $AA^T$ is of the form:
  $$\begin{pmatrix}
0 & * & * & \cdots & * \\
* & 0 & * & \cdots & * \\
* & * & 0 & \cdots & * \\
\vdots & \vdots & \vdots & \cdots & \vdots \\
* & * & * & \cdots & 0
\end{pmatrix}$$
Then the number of such matrices $A$ is:
A) $(4^3)^{10}$  B)$(4^2)^{10}$  C)$4^{10}$  D) $1$

Since all the diagonal elements of $AA^T$ is zero I could realize that it is a skew-symmetric matrix. But, I'm no able to understand how I can use this result for finding the possibilities of the original matrix. I would like hints rather than answers.

REPLY [2 votes]: Since you mentioned symmetry, try simplifying the expression $(AA^T)^T$. What do you get, and what does this say about $AA^T$?
To find the possibilities of the original matrix, try writing the $(i,i)$-th diagonal element of $AA^T$ in terms of $A$, then equate the result with $0$.