Mathematics - Matrices Question with Solution | TestHub

The number of matrices for which $p$ does not divide $\mathrm{Tr}(A)=(p-1)p^2$ of these $(p-1{)}^2$ are such that $p$ divides $|A|$. The number of matrices for which $p$ divides $\mathrm{Tr}(A)$ and $p$ does not divides $|A|$ are $(p-1{)}^2$ $$\begin{array}{rl}& \therefore \text{ Required number }\\ & =(p-1)p^2-(p-1{)}^2+(p-1{)}^2\\ & =p^3-p^2\end{array}$$