3-by-3 determinants

| \begin{matrix}a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{matrix} |
= a_{11} a_{22} a_{33} + a_{12} a_{23} a_{31} + a_{13} a_{21} a_{32}

− a_{11} a_{23} a_{32} − a_{12} a_{21} a_{33} − a_{13} a_{22} a_{31}

| \begin{matrix}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{matrix} |
= a_{11} a_{22} a_{33} a_{44} + a_{11} a_{23} a_{34} a_{42}

+ a_{11} a_{24} a_{32} a_{43} + a_{12} a_{21} a_{34} a_{43} + a_{12} a_{23} a_{31} a_{44} + a_{12} a_{24} a_{33} a_{41} + a_{13} a_{21} a_{32} a_{44} + a_{13} a_{22} a_{34} a_{41} + a_{13} a_{24} a_{31} a_{42} + a_{14} a_{21} a_{33} a_{42} + a_{14} a_{22} a_{31} a_{43} + a_{14} a_{23} a_{32} a_{41} − a_{11} a_{22} a_{34} a_{43} − a_{11} a_{23} a_{32} a_{44} − a_{11} a_{24} a_{33} a_{42} − a_{12} a_{21} a_{33} a_{44} − a_{12} a_{23} a_{34} a_{41} − a_{12} a_{24} a_{31} a_{43} − a_{13} a_{21} a_{34} a_{42} − a_{13} a_{22} a_{31} a_{44} − a_{13} a_{24} a_{32} a_{41} − a_{14} a_{21} a_{32} a_{43} − a_{14} a_{22} a_{33} a_{41} − a_{14} a_{23} a_{31} a_{42}