Why LU decomposition is not used to give elementary proof of $\det (AB) = \det (A) \det (B) $ for square matrices?
It seems for any square matrix $ A $, $ A = PLU$
P is permutation matrix, L lower triangular matrix, U upper triangular matrix.
Using elementary row operations :
$ \det (PB) = \det (B) \det (P) $,
$\det (LB) = \det (L) \det (B) $,
$\det (UB) = \det (U) \det (B) $
Regarding the last 2 identities: You can obtain $ LB $ by performing row additions to the matrix $ G=TB $, where $ T $ is a diagonal matrix such that $ T_{ii} = L_{ii} $.
Starting from the last row, take each row of $ G $ and add multiples of rows above it to obtain the rows of $ LB $, this implies $ det (LB) = det (TB) =det (T) det (B)= det (L) det (B) $
Similar trick works for $ UB $ by adding multiples of rows below a given row starting from the first row.
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