For $A\in\mathbb F^{n\times n}$, $\det A=\sum_{\sigma\in S_n}\operatorname{sgn}(\sigma)\prod_{i=1}^n a_{i,\sigma(i)}$. Each term selects one entry from every row and column. The permutation sign records the parity of crossings. This formula proves existence of the determinant but has $n!$ terms, so it is not a practical large-matrix algorithm.
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