I don't think that this one has been asked, but I've been told that there is proof for a fundamental property of determinants that hinges on LDU decomposition of a matrix. I just can't figure out what property that is and I don't even know why one would want to do a LDU decomposition. I only know about what LDU decomposition is. Please help me.
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Here is a guess. When I was a freshman, they taught me a proof of the fact that $\det AB = \det A \cdot \det B$ that used elementary matrices. I can imagine a similar proof using LDU decomposition.
I mean, suppose that you already know that each matrix has such a decomposition. Suppose that you've also established that multiplying a column of a matrix by a number multiplies the determinant by the same number. Suppose that it's already known that adding one column or row to another doesn't change the determinant.
Using this knowledge, it is very easy to prove that $\det AB = \det A \cdot \det B$ when $B$ is either a diagonal, a unit upper-triangular or a unit lower-triangular matrix. It is also quite easy to establish that $\det LDU = \det D$.
And knowing all this, we can easily prove the equality in general. If $B=LDU$, then $$ \det AB = \det ALDU = \det ALD = \det AL \cdot \det D = \det A \cdot \det D = \det A \cdot \det B. $$
But again, this is just a guess.