已知4阶行列式D=|1& 2& 1& 1 2& 3& (2)^2& (2)^3 3& 4& (3)^2& (3)^3 4& 1& (4)^2& (4)^3(i=1,2,3,4)为D中第i行，第2列元素的代数余子式.

已知$4$阶行列式$D=\left|\begin{array}{llll}1& 2& 1& 1\\ 2& 3& {2}^{2}& {2}^{3}\\ 3& 4& {3}^{2}& {3}^{3}\\ 4& 1& {4}^{2}& {4}^{3}\end{array}\right|$，求${A}_{12}+{A}_{22}+{A}_{32}+{A}_{42}$，其中${A}_{i2}\left(i=1,2,3,4\right)$为$D$中第$i$行，第$2$列元素的代数余子式.