> [!NOTE] Theorem (Proof of existence in the general case) > For any $A\in \mathbb{F}^{n\times n },$ we define a scalar $\det A$ by the formula $\det A = \sum_{\sigma\in S_{n}} \text{sgn }(\sigma) a_{1\sigma(1)} a_{2\sigma(2)}\dots a_{n\sigma(n)}$that satisfies the properties of the definition of a determinant function, where $S_{n}$ is the permutation group on $n$ symbols. >*Proof*.