The number of in such that the trace of is not divisible by but is divisible by is [Note: The trace of a matrix is the sum of its diagonal entries.] [2010]

Question

Let $p$ be an odd prime number and $T_{p}$ be the following set of $2 \times 2$ matrices:

$T_{p} = {A = [\begin{matrix} a & b \\ c & a \end{matrix}] : a, b, c \in {0, 1, 2, \dots, p - 1}}$ [2010]

Q. The number of $A$ in $T_{p}$ such that the trace of $A$ is not divisible by $p$ but $\det (A)$ is divisible by $p$ is

[Note: The trace of a matrix is the sum of its diagonal entries.]

Smart Achievers · Accepted Answer

(3)

Trace of $T_{p}$ , will not be divisible by $p$ , if $a \neq 0 .$

$| A |$ $= a^{2} - b c,$ for $(a^{2} - b c)$ to be divisible by $p$ , there are exactly $(p - 1)$ ordered pairs $(b, c)$ for any value of $a$ .

$∴$ Required number is ${(p - 1)}^{2}$

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