Question

The number of $A$ in $T_p$ such that det $\left(A\right)$ is not divisible by $p$ is

• A. $p^3-5p$
• B. $p^3-p^2$
• C. $2p^2$
• D. $p^3-3p$

Answer 1

The correct option is B $p^3-p^2$

The number of matrices for which p does not divide $Tr\left(A\right)$ is $(p–1)p^2$ of these $(p–1{)}^2$are such that $p||A|$. The number of matrices for which $p$ divides $Tr\left(A\right)$ and $p$ does not divide $|A|$ are $(p–1{)}^2$.
$\therefore$ Required number is $(p–1)p^2–(p–1{)}^2+(p–1{)}^2=p^3–p^2$