I have been reading Linear Algebra Done Wrong and came across this Lemma:
Lemma 3.6. For a square matrix A and an elementary matrix E (of the same size)
det(AE) = (det A)(det E)
Proof: The proof can be done just by direct checking: determinants of special matrices are easy to compute; right multiplication by an elementary matrix is a column operation, and effect of column operations on the determinant is well known.
This can look like a lucky coincidence, that the determinants of elementary matrices agree with the corresponding column operations, but it is not a coincidence at all.
Namely, for a column operation the corresponding elementary matrix can be obtained from the identity matrix I by this column operation. So, its determinant is 1 (determinant of I) times the effect of the column operation. And that is all! It may be hard to realize at first, but the above paragraph is a complete and rigorous proof of the lemma!
I am having a very hard time understanding this proof any help would be great, thanks!