Question

The number of $A$ in $T_p$ such that $A$ is either symmetric or skew-symmetric or both, and $\text{det}\left(A\right)$ divisible by $p$ is

• A. $(p-1{)}^2$
• B. $2p-1$
• C. $(p-1{)}^2+1$
• D. $2(p-1)$

Answer 1

The correct option is B $2p-1$

If $A$ is skew-symmetric, $|A|=–b^2$
$\therefore |A|$ if $b=0$ in which case $A$ is symmetric.
If $A$ is $\left[\begin{array}{cc}a& b\\ b& a\end{array}\right],|A|=a^2-b^2$
$\therefore p|A|$ if either $p|a+b\text{or}p|a–b$.
The second case occurs only when $a=b$ which gives $p$ choices. The first case again gives $p$ choices with $a=0=b$ counted again.
$\therefore$ The number of such matrices is $2p–1$.