# Elementary proof of multiplicative property of determinant

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.

• It is not entirely clear to me why should $\det(LB)=\det L\det B$ or $\det (UB)=\det U\det B$ – Gae. S. Jul 12 at 15:56
• And how is it usually proved? The proof we were given in the class was basically a repeat of LU proof. (I guess it is always useful repeat such important stuff for students). But i personally love to think just of $\mathrm{det}$ as of constant by which volumes gets multiplied, then (at least for $\mathbb{R}^{n}$) this formula becomes obvious, and if we wish to extend it for other commutative rings, there is an algebraic trick. – Rybin Dmitry Jul 12 at 15:58
• Well, I wish I knew a proper answer which I could share with you, but I suspect that a proof involving LU decomposition might be considered non-elementary. There is no proof more elementary than one which uses little more than the definition of a determinant. See chapter 8 of math.ucdavis.edu/~anne/linear_algebra/mat67_course_notes.pdf and chapter 6 of cip.ifi.lmu.de/~grinberg/primes2015/…. – Thomas Winckelman Jul 12 at 16:02
• The usual intro proof uses elementary matrices/operations, which you would need anyway to prove your three identities. Seems like you're inserting a middle-man. – Randall Jul 12 at 16:03
• I agree with @Randall – Thomas Winckelman Jul 12 at 16:05