${C_2} \to {C_2} - {C_1},{C_3} \to {C_3} - {C_1}$

$ \Rightarrow \left| {\begin{array}{*{20}{c}}
1&0&0\\
2&{b - 2}&{c - 2}\\
4&{{b^2} - 4}&{{c^2} - 4}
\end{array}} \right|$

$ = \left( {b - 2} \right)\left( {c - 2} \right)\left| {\begin{array}{*{20}{c}}
1&1\\
{b + 2}&{c + 2}
\end{array}} \right|$

$\left| A \right| = \left( {b - 2} \right)\left( {c - 2} \right)\left( {c - b} \right)$

$2,b,c$ are in $AP \Rightarrow 2,2 + d,2 + 2d$

$ \Rightarrow \left| A \right| = \left( d \right)\left( {2d} \right)\left( d \right) = 2{d^3} \in \left[ {2,16} \right]$

$ \Rightarrow {d^3} \in \left[ {1,8} \right]$

$ \Rightarrow 2d \in \left[ {2,4} \right]$

$ \Rightarrow 2 + 2d \in \left[ {4,6} \right]$