Let $a \in R$ and A be a matrix of order $3 \times 3$ such that $\det (A) = - 4$ and $A + I = \left[ {\begin{array}{*{20}{l}} 1&{{\rm{ a }}}&1\\ 2&1&0\\ a&1&2 \end{array}} \right]$ , where I is the identity matrix of order $3 \times 3$. If det $((a + 1){\mathop{\rm adj}\nolimits} ((a - 1)A))$ is ${2^m}{3^n},m$, $n \in \{ 0,1,2, \ldots ,20\} $, then $m + n$ is equal to :