I am looking for a non-permutational definition of determinant. The definition should have these properties:
1: Calculational power (easily applicable, it cold be used for practical calculations).
2: It should hold the usual properties (otherwise it wouldn't be a determinant, right!)
3: No permutation, No permutation, please no permutations.
I would also appreciate if you apply the definition and calculate the determinant of a $4×4$ matrix.
For all I care the definition could be from an obsolete parchment, but it needs to have those three properties.
Appreciate all the help!
For the $3\times 3$-determinant we can use the Rule of Sarrus. Then the $4\times 4$ determinant reduces to the $3\times 3$ determinant, because of $${\begin{vmatrix}a&b&c&d\\e&f&g&h\\i&j&k&l\\m&n&o&p\end{vmatrix}}=a\,{\begin{vmatrix}f&g&h\\j&k&l\\n&o&p\end{vmatrix}}-b\,{\begin{vmatrix}e&g&h\\i&k&l\\m&o&p\end{vmatrix}}+c\,{\begin{vmatrix}e&f&h\\i&j&l\\m&n&p\end{vmatrix}}-d\,{\begin{vmatrix}e&f&g\\i&j&k\\m&n&o\end{vmatrix}}.$$