Paragraph: Let p be an odd prime number and T_p be the following...

Given, A=\left[$\begin{array}{ll}a & b \\ c & a\end{array}$\right], $a, b, c \in\{0,1,2, \ldots, p-1\}$ If $A$ is skew-symmetric matrix, then $a=0, b=-c$ $\therefore \quad|A|=-b^2$. Thus, $P$ divides $|A|$ only when $b=0$...(i) Again, if $A$ is symmetric matrix, then $b=c$ and $|A|=a^2-b^2$. Thus, $p$ divides $|A|$ if either $p$ divides $(a-b)$ or $p$ divides $(a+b)$. $p$ divides $(a-b)$, only when $a=b$ ie, $a=b \in\{0,1,2, \ldots,(p-1)\}$ ie, pchoices $p$ divides $(a+b)$. $\Rightarrow p$ choices, including $a=b=0$ included in (i) $\therefore$ Total number of choices are $(p+p-1)=2 p-1$. Hence, (c) is the correct option.