> [!NOTE] Proposition (Proof of existence for $n=3$) > For $A\in \mathbb{R}^{3\times{3}},$ define $\det A = \sum_{\sigma\in S_{3}}\text{sgn} (\sigma)a_{1\sigma(1)} a_{2\sigma(2)} a_{3\sigma(3)}$ >Proof. Note that $S_{3} = \{ \text{id}, (1,2), (1,3),(2,3), (1,2,3), (1,3,2) \}$so $\det A=a_{11}a_{22}a_{33}-a_{12}a_{21}a_{33}-a_{13}a_{22}a_{31}-a_{11}a_{23}a_{32}+a_{12}a_{23}a_{31}+a_{13}a_{21}a_{32}$hence $\det A=a_{11}\det\begin{pmatrix}a_{22}&a_{23}\\a_{32}&a_{33}\end{pmatrix}-a_{12}\det\begin{pmatrix}a_{21}&a_{23}\\a_{31}&a_{33}\end{pmatrix}+a_{13}\det\begin{pmatrix}a_{21}&a_{22}\\a_{31}&a_{32}\end{pmatrix}.$Now it is clear that the properties of the definition hold for this function.