\[\begin{equation} \tag{18} \operatorname{det}(\mathbf{A})=\prod_{i} \lambda_{i} \quad \lambda_{i}=\operatorname{eig}(\mathbf{A}) \end{equation}\]
\[\begin{equation} \tag{19} \operatorname{det}(c \mathbf{A})=c^{n} \operatorname{det}(\mathbf{A}), \quad \text { if } \mathbf{A} \in \mathbb{R}^{n \times n} \end{equation}\]
\[\begin{equation} \tag{20} \operatorname{det}\left(\mathbf{A}^{T}\right)=\operatorname{det}(\mathbf{A}) \end{equation}\]
\[\begin{equation} \tag{21} \operatorname{det}(\mathbf{A B})=\operatorname{det}(\mathbf{A}) \operatorname{det}(\mathbf{B}) \end{equation}\]
The determinant of a tranformation matrix is the scale of area/volume of the shape before and after the tranformation. \(\mathbf{A B}\) are two consecutive transformations, therefore its determinant is the product of two scales.
\[\begin{equation} \tag{22} \operatorname{det}\left(\mathbf{A}^{-1}\right)=1 / \operatorname{det}(\mathbf{A}) \end{equation}\]
\[\begin{equation} \tag{23} \operatorname{det}\left(\mathbf{A}^{n}\right)=\operatorname{det}(\mathbf{A})^{n} \end{equation}\]
\[\begin{equation} \tag{24} \operatorname{det}\left(\mathbf{I}+\mathbf{u v}^{T}\right)=1+\mathbf{u}^{T} \mathbf{v} \end{equation}\]
\[\begin{equation} \tag{25} \begin{array}{l}{\text { For } n=2:} \\ {\qquad \operatorname{det}(\mathbf{I}+\mathbf{A})=1+\operatorname{det}(\mathbf{A})+\operatorname{Tr}(\mathbf{A})}\end{array} \end{equation}\]
\[\begin{equation} \tag{26} \begin{array}{l}{\text { For } n=3:} \\ {\qquad \operatorname{det}(\mathbf{I}+\mathbf{A})=1+\operatorname{det}(\mathbf{A})+\operatorname{Tr}(\mathbf{A})+\frac{1}{2} \operatorname{Tr}(\mathbf{A})^{2}-\frac{1}{2} \operatorname{Tr}\left(\mathbf{A}^{2}\right)}\end{array} \end{equation}\]