$\det A={\displaystyle \sum _{\sigma }{(-1)}^{\epsilon (\sigma )}{A}_{1\sigma (1)}}{A}_{2\sigma (2)}\mathrm{...}{A}_{n\sigma (n)}$

where σ = (σ(1), σ(2), …, σ(n)) is a permutation of the subscripts (1, 2, …, n), ε(σ) is the number of inversions in permutation σ, and the sum denoted by Σ is for all permutations

Note 1 to entry: The determinant of a matrix is equal to the determinant of the n vectors, the coordinates of which are the elements of the rows or of the columns.

Note 2 to entry: The determinant of the matrix $A=\left(\begin{array}{ccc}{A}_{11}& \cdots & A_{1n}\\ ⋮& \ddots & ⋮\\ A_{n1}& \cdots & A_{nn}\end{array}\right)$ is denoted A or $\left|\begin{array}{ccc}{A}_{11}& \cdots & A_{1n}\\ ⋮& \ddots & ⋮\\ A_{n1}& \cdots & A_{nn}\end{array}\right|$.