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

Answer 1

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

$a$ cannot be equal to 0 as the trace would be divisible by $p$.
Hence, $a$ can acquire values from $\{1,2,...,p-1\}$
Now let us select some value of $a$ from the given set of values.
Let the remainder of $a^2$ be $r$ when divided by $p$.
Now, $a^2-bc$ must be divisible by $p$. Hence, $bc$ must have the remainder $r$ when divided by $p$.
Now, $b$ or $c$ cannot be 0.
If we select some value of $b$ from $\{1,2,...,n-1\}$ , we can exactly have one value of $c$ such that the remainder of $bc$ is r.
We can select $b$ in $p-1$ ways.
For every value of $a$, $b$ can be selected in $p-1$ ways.
$a$ can be selected in $p-1$ ways.
Hence, total number of possible matrices $=(p-1)\times (p-1)$