Your condition comes down to asking for $T_1(x,x)=T_2(x,x)$ for all $x$ --- there seems to be no condition on $T_i(x,y)$ for $x\ne y$. So for each $x$, define $T_1(x,x)$ to be anything you like, then define $T_2(x,x)$ to be equal to $T_1(x,x)$, and then define $T_i(x,y)$ any way you like for $x\ne y$.
It seems to me that despite the tags there is no linear algebra nor any modular arithmetic here. Perhaps you have left something out.
EDIT: The problem has been edited into something completely different, but not at all clear.
"$T_1(x,y)=T_2(x,y)$" is usually taken to mean equality for all pairs $(x,y)$ in $X\times X$, but you write "$T_1(x,y)=T_2(x,y)$, but not for all values from $X\times X$." So it's not at all clear for which values you want to have $T_1(x,y)=T_2(x,y)$. Please think about what you really mean, and write something that I can comprehend.
Also, I still see no justification for the tags used.