Question

The number of $A$ in $T_p$ such that the trace of $A$ is not divisible by $p$ but det $\left(A\right)$ is divisible by $p$ is
[Note : The trace of a matrix is the sum of its diagonal entries]

• A. $(p-1{)}^2$
• B. $p^3-(p-1{)}^2$
• C. $(p-1)(p^2-2)$
• D. $(p–1)(p^2–p+1)$

Answer 1

The correct option is A $(p-1{)}^2$

$Tr\left(A\right)=2a$ will be divisible by $p$ if and only if $a=0$.
We want to find all matrices $\left[\begin{array}{cc}a& b\\ c& a\end{array}\right]$ with $a=1,2,.....,p–1$ for which $a^2–bc=0$ (mod $p$).
There are exactly $p–1$ ordered pairs $(b,c)$ for any value of $a$.
$\therefore$ Required number is $(p–1{)}^2$