Question

Read the following information carefully and answer the following request. Let $p$ be an odd prime number and $T_p$ be the following set of $2\times 2$ matrices. The number of $A$ in $T_p$ such that $A$ is either symmetric or skew-symmetric or both and $det\left(A\right)$ is divisible by $p$ is

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

Answer 1

The correct option is D

$2p-1$

Explanation for the correct option:

Let us assume the matrix,

$$\begin{gathered} T_p=A=\left[\right(a,b),(c,a\left)\right]:a,b,c\in 0,1,....,p-1 \\ A=\left[\begin{array}{cc}a& b\\ c& a\end{array}\right] \\ det\left(A\right)=\left|\begin{array}{cc}a& b\\ c& a\end{array}\right| \\ =a^2-bc \end{gathered}$$

If $A$ is symmetric then $b=c$

so, $\left|A\right|=a^2-b^2$ which is divisible by $p$ if $\left(a+b\right)$ or $\left(a-b\right)$ is divisible by $p$

Now, $\left(a+b\right)$ is divisible by $p$ if $\left(a,b\right)$ can take values $(1,p-1),(2,p-2),(3,p-3)......(p-1,1)$ so total number of ways are $\left(p-1\right)$

Similarly, for $\left(a-b\right)$ possible values of $\left(a,b\right)$ are $(0,0),(1,1),(2,2)....(p-1,p-1)$ making the total number of ways $p$

So, Total number of ways are $$\begin{gathered} =p+p-1 \\ =2p-1 \end{gathered}$$

Hence, the correct option is (D)