Properties of Determinants

For any $𝐴 \in 𝐹^{𝑛 \times 𝑛}$ , $\det (𝐴^{𝖳}) = \det (𝐴)$ .

For any $𝐴, 𝐵 \in 𝐹^{𝑛 \times 𝑛}$ , $\det (𝐴 𝐵) = \det (𝐴) \det (𝐵)$ .

Let $𝐴 \in 𝐹^{𝑛 \times 𝑛}$ . Then $𝐴$ is invertible if and only if $\det (𝐴) \neq 0$ , in which case:

\det (𝐴^{- 1}) = \frac{1}{\det (𝐴)} .