MHB Which Permutations Should Be Excluded in Leibniz Formula for Determinant?

Amer
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Hey
libniz formula
i can't post picture and i can't use latex :(

if we want to use Libniz forumla for find the determinant of A such that

A = 1..-1..0..3
...2..-3..0..4
...0..2..-1..5
...1..2..2..3

sigma here from S_4 which has 4! elements which element will we choose
 
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We must exclude permutations which map 1 to 3 or 2 to 3 or 3 to 1. Suppose 1 is mapped to 1. Then 2 is mapped to 2 or 4. Therefore, the permutations that start with 1 are (1, 2, 3, 4), (1, 2, 4, 3), (1, 4, 2, 3) and (1, 4, 3, 2). Now try to come up with the remaining 6 permutations.
 
The world of 2\times 2 complex matrices is very colorful. They form a Banach-algebra, they act on spinors, they contain the quaternions, SU(2), su(2), SL(2,\mathbb C), sl(2,\mathbb C). Furthermore, with the determinant as Euclidean or pseudo-Euclidean norm, isu(2) is a 3-dimensional Euclidean space, \mathbb RI\oplus isu(2) is a Minkowski space with signature (1,3), i\mathbb RI\oplus su(2) is a Minkowski space with signature (3,1), SU(2) is the double cover of SO(3), sl(2,\mathbb C) is the...

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