Determinant of a product
The derivation is very similar to the proof of the Leibniz rule.
Consider matrices and let denote the th row of . Row of the product obviously, can be written as Using Property V times, we have
By analogy with permutation matrices denote
Recall the link between and
If we had proved the multiplication rule for (2), we would have
In the theory of permutations (3) is proved without relying on the multiplication rule. I am going to use (3) as a shortcut that explains the idea. Combining (1) and (3) we obtain by the Leibniz formula