Property of $10\times 10 $ matrix Let $A$ be a $10 \times 10$ matrix such that each entry is either $1$ or $-1$. Is it true that $\det(A)$ is divisible by $2^9$?

Alternatively, we can choose $A$ as below:
$$A = {\left[ {\begin{array}{*{20}{c}}
  1&1& \ldots &1 \\ 
  { - 1}&1& \ldots &1 \\ 
  { - 1}&{ - 1}& \ldots &{ - 1} \\ 
   \vdots &{}&{}&{} \\ 
  { - 1}&{ - 1}& \ldots &{ - 1} 
\end{array}} \right]_{10 \times 10}}$$
If we add fist row to another rows, we get:
$$A = {\left[ {\begin{array}{*{20}{c}}
  1&1& \ldots & \ldots &1 \\ 
  0&2& \ldots & \ldots &2 \\ 
  0&0& \ldots & \ldots &2 \\ 
   \vdots &{}&{}&{}&{} \\ 
  0&0& \ldots &0&2 
\end{array}} \right]_{10 \times 10}}\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\det (A) = {2^9}$$
