Question

Let $p$ be an odd prime number and $T _{p}$ be the following set of $2 \times 2$ matrices :
$T_{p}=\left\{A=\begin{bmatrix}a & b \\ c & a\end{bmatrix}: a\right.$, b.c $\left.\in\{0.1,2 . \ldots, p-1\}\right\}$
The number of $A$ in $T _{ p }$ such that $A$ is either symmetric or skew-symmetric or both, and det $(A)$ divisible by $p$ is

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

Answer 1

Answer: (D)

We must have $a^{2}-b^{2}=k p$
$\Rightarrow (a+b)(a-b)=k p$
$\Rightarrow $ either $a - b =0$ or $a + b$ is a multiple of $p$
and when $a + b =$ multiple of $p$
$ \Rightarrow a , b$ has $p -1$
$\therefore $ Total number of matrices $= p + p -1$
$=2 p -1$