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
(1)
Here by Property III
if among rows there are equal vectors. The remaining matrices with nonzero determinants are permutation matrices, so
(2)
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).
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