Determinant of the unit matrix proof

Question

I need to prove that the determinant of any unit matrix is 1, using the defition of the sign of permutation. As far as I know the sign is defined as +1 if the permutation is even, and -1 if the permutation is odd.

Answer 1

In the formula for the determinant of a matrix $(a_{ij})_{1\le i,j\le n}$: $$\sum_{\sigma\in\mathfrak S_n}(-1)^{\varepsilon(\sigma)}a_{1\mkern1.5mu\sigma(1)}a_{2\mkern1.5mu\sigma(2)}\dots a_{n\mkern1.5mu\sigma(n)},$$ the only permutation for which all factors are non zero (and equal to $1$) is the identity, which is an even permutation.