Question

The number of A in T${}_p$ such that det(A) is not divisible by p is?

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

Answer 1

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

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