Let p be an odd prime number and $T_p$ be the following set of 2 × 2 matrices $T_p=\left\{A=\begin{bmatrix}a&b\\c&a\end{bmatrix}: a, b, c ∈ \{0, 1, 2,..., p-1\}\right\}$
The number of A in $T_p$ such that the trace of A is not divisible by p is
Answer & explanation
Correct answer: option 4
We have,
Total number of matrices in $T_p =p^3$
Total number of matrices in $T_p$ whose trace is divisible by p is $p^2$ [∵ a = 0 and each one of b and c can take p values]
Total number of matrices whose trace is not divisible by p is $p^3 -p^2 = p^2 (p-1)$.
Out of these $(p-1)^2$ matrices are such that p divides det (A).
The number of matrices for which p divides trace of A and p does not divide det (A) is $(p-1)^2$.
∴ Required number of matrices = $(p-1) p^2 - (p-1)^2 + (p-1)^2 =p^3 - p^2$