$$ \left| \begin{array}{llll} a_{11} & a_{12} & a_{13} & a_{14}\\ a_{21} & a_{22} & a_{23} & a_{24} \\ a_{31} & a_{32} & a_{33} & a_{34} \\ a_{41} & a_{42} & a_{43} & a_{44} \end{array} \right|= $$ $$ \begin{array}{ccc} &=&a_{11}a_{22}a_{33}a_{44} - a_{11}a_{22}a_{34}a_{43} - a_{11}a_{32}a_{23}a_{44} + a_{11}a_{32}a_{24}a_{43} + a_{11}a_{42}a_{23}a_{34} - a_{11}a_{42}a_{24}a_{33} - \\ &-& a_{21}a_{12}a_{33}a_{44} + a_{21}a_{12}a_{34}a_{43}+ a_{21}a_{32}a_{13}a_{44} - a_{21}a_{32}a_{14}a_{43} - a_{21}a_{42}a_{13}a_{34} + a_{21}a_{42}a_{14}a_{33}+ \\ &+& a_{31}a_{12}a_{23}a_{44} - a_{31}a_{12}a_{24}a_{43} - a_{31}a_{22}a_{13}a_{44} + a_{31}a_{22}a_{14}a_{43} + a_{31}a_{42}a_{13}a_{24} - a_{31}a_{42}a_{14}a_{23} - \\ &-& a_{41}a_{12}a_{23}a_{34} + a_{41}a_{12}a_{24}a_{33}+ a_{41}a_{22}a_{13}a_{34} - a_{41}a_{22}a_{14}a_{33} - a_{41}a_{32}a_{13}a_{24} + a_{41}a_{32}a_{14}a_{23} \ . \end{array} $$

Разложение определителя по первому столбцу $$ =a_{11} \left| \begin{array}{lll} a_{22} & a_{23} & a_{24} \\ a_{32} & a_{33} & a_{34} \\ a_{42} & a_{43} & a_{44} \end{array} \right| -a_{21} \left| \begin{array}{lll} a_{12} & a_{13} & a_{14}\\ a_{32} & a_{33} & a_{34} \\ a_{42} & a_{43} & a_{44} \end{array} \right| + a_{31} \left| \begin{array}{lll} a_{12} & a_{13} & a_{14}\\ a_{22} & a_{23} & a_{24} \\ a_{42} & a_{43} & a_{44} \end{array} \right| - a_{41} \left| \begin{array}{lll} a_{12} & a_{13} & a_{14}\\ a_{22} & a_{23} & a_{24} \\ a_{32} & a_{33} & a_{34} \end{array} \right| $$ Тождество (равенство) Сильвестра: $$ \left| \begin{array}{llll} a_{11} & a_{12} & a_{13} & a_{14}\\ a_{21} & a_{22} & a_{23} & a_{24} \\ a_{31} & a_{32} & a_{33} & a_{34} \\ a_{41} & a_{42} & a_{43} & a_{44} \end{array} \right|\cdot \left| \begin{array}{ll} a_{22} & a_{23} \\ a_{32} & a_{33} \end{array} \right| = $$ $$ = \left| \begin{array}{lll} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{array} \right| \cdot \left| \begin{array}{lll} a_{22} & a_{23} & a_{24} \\ a_{32} & a_{33} & a_{34} \\ a_{42} & a_{43} & a_{44} \end{array} \right| - \left| \begin{array}{llll} a_{12} & a_{13} & a_{14}\\ a_{22} & a_{23} & a_{24} \\ a_{32} & a_{33} & a_{34} \end{array} \right| \cdot \left| \begin{array}{llll} a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \\ a_{41} & a_{42} & a_{43} \end{array} \right| $$