Rule for calculating determinant
This is one of those cases when calculations explain the result.
Let denote the th unit row-vector. Row obviously, can be decomposed as Recall that by Property V, the determinant of is a multilinear function of its rows. Using Property V times, we have
Here by Property III
if among rows there are equal vectors. The remaining matrices with nonzero determinants are permutation matrices, so
Different-rows-different-columns rule. Take a good look at what the condition implies about the location of the factors of the product The rows to which the factors belong are obviously different. The columns are also different by the definition of a permutation matrix. Conversely, consider any combination of elements such that no two elements belong to the same row or column. Rearrange the first indices in an ascending order, from to This leads to a renumbering of the second indices. The product becomes Since the original second indices were all different, the new ones will be too. Hence, and such a term must be in (2).
This rule alternatively can be called a cross-out rule because in practice it is used like this: take, say, element and cross out the row and column it belongs to. The next factor is selected from the remaining matrix. In that matrix, again cross out the row and column the second factor sits in. Continue like this until you have factors. Multiply the resulting product by the determinant of an appropriate permutation matrix, and you have one term of the Leibniz formula.
Remark 1. (2) is the Leibniz formula. The formula at the right of (1) is the Levy-Civita formula. The difference is that the Levy-Civita formula contains many more zero terms, while in (2) a term can be zero only if .
Remark 2. Many textbooks instead of write in (2) signatures of permutations. Using is better because a) you save time by skipping the theory of permutations and b) you need a rule to calculate signatures of permutations and is such a rule (see an example).