Let p be an odd prime number and $\mathrm { T } _ { \mathrm { p } }$ be the following set of $2 \times 2$ matrices: $$\mathrm { T } _ { \mathrm { p } } = \left\{ \mathrm { A } = \left[ \begin{array} { l l } \mathrm { a } & \mathrm {~b} \\ \mathrm { c } & \mathrm { a } \end{array} \right] : \mathrm { a } , \mathrm {~b} , \mathrm { c } \in \{ 0,1,2 , \ldots , \mathrm { p } - 1 \} \right\}$$ The number of $A$ in $T _ { p }$ such that the trace of $A$ is not divisible by $p$ but $\operatorname { det } ( A )$ is divisible by $p$ is [Note : The trace of a matrix is the sum of its diagonal entries.] A) $( \mathrm { p } - 1 ) \left( \mathrm { p } ^ { 2 } - \mathrm { p } + 1 \right)$ B) $\mathrm { p } ^ { 3 } - ( \mathrm { p } - 1 ) ^ { 2 }$ C) $( p - 1 ) ^ { 2 }$ D) $( \mathrm { p } - 1 ) \left( \mathrm { p } ^ { 2 } - 2 \right)$
Let p be an odd prime number and $\mathrm { T } _ { \mathrm { p } }$ be the following set of $2 \times 2$ matrices:
$$\mathrm { T } _ { \mathrm { p } } = \left\{ \mathrm { A } = \left[ \begin{array} { l l } \mathrm { a } & \mathrm {~b} \\ \mathrm { c } & \mathrm { a } \end{array} \right] : \mathrm { a } , \mathrm {~b} , \mathrm { c } \in \{ 0,1,2 , \ldots , \mathrm { p } - 1 \} \right\}$$
The number of $A$ in $T _ { p }$ such that the trace of $A$ is not divisible by $p$ but $\operatorname { det } ( A )$ is divisible by $p$ is\\
[Note : The trace of a matrix is the sum of its diagonal entries.]\\
A) $( \mathrm { p } - 1 ) \left( \mathrm { p } ^ { 2 } - \mathrm { p } + 1 \right)$\\
B) $\mathrm { p } ^ { 3 } - ( \mathrm { p } - 1 ) ^ { 2 }$\\
C) $( p - 1 ) ^ { 2 }$\\
D) $( \mathrm { p } - 1 ) \left( \mathrm { p } ^ { 2 } - 2 \right)$