Mathematics - Matrices Question with Solution | TestHub

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