\begin{align}
&\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| \\\\
= &{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} \\\\
= &{a}_{11}\left( {{a}_{22}{a}_{33} - {a}_{23}{a}_{32}}\right) \\
&+ {a}_{12}\left( {{a}_{23}{a}_{31} - {a}_{21}{a}_{33}}\right) \\
&+ {a}_{13}\left( {{a}_{21}{a}_{32} - {a}_{22}{a}_{31}}\right) \\\\
= &{a}_{11}\left| \begin{array}{ll} {a}_{22} & {a}_{23} \\ {a}_{32} & {a}_{33} \end{array}\right| - {a}_{12}\left| \begin{array}{ll} {a}_{21} & {a}_{23} \\ {a}_{31} & {a}_{33} \end{array}\right| + {a}_{13}\left| \begin{array}{ll} {a}_{21} & {a}_{22} \\ {a}_{31} & {a}_{32} \end{array}\right| .
\end{align}$$
由此可知, 一个三阶行列式可由三个二阶行列式表示,
且其系数全部来自于原行列式的第一行.