# How do I test two square matrices are transpose to each other if only the column vector summations are known?

Given two secret square matrices, say $$\left( {\begin{array}{*{20}{c}} {{a_{11}}}&{{a_{12}}}&{{a_{13}}}\ {{a_{21}}}&{{a_{22}}}&{{a_{23}}}\ {{a_{31}}}&{{a_{32}}}&{{a_{33}}} \end{array}} \right)$$ and its transpose $$\left( {\begin{array}{*{20}{c}} {{a_{11}}}&{{a_{21}}}&{{a_{31}}}\ {{a_{12}}}&{{a_{22}}}&{{a_{32}}}\ {{a_{13}}}&{{a_{23}}}&{{a_{33}}} \end{array}} \right)$$, suppose the summations of the column vectors of both matrices, i.e.,$$\sum\limits_{i = 1}^3 {{a_{ij}}} ,j \in \left[ {1,3} \right]$$ and $$\sum\limits_{j = 1}^3 {{a_{ij}}} ,i \in \left[ {1,3} \right]$$ are public. I wonder whether there exists an algorithm, which is allowed to query a constant (i.e., independent of the dimension of the square matrix) number of positions in both matrices, such that it can determine whether these two secret matrices are transpose to each other with non-neglibile probability?