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Is there a easy proof for $\det(EA)=\det(E)\det(A)$, where E is an elementary matrix? Determinant here is defined through cofactor expansion.

I checked Proof $\det(AB)=\det(A)\det(B)$ and one answer there linked a wiki - https://proofwiki.org/wiki/Determinant_with_Row_Multiplied_by_Constant

In proof 1 section of the link, they put, enter image description here However, when I go to "Determinant of Matrix Product" - https://proofwiki.org/wiki/Determinant_of_Matrix_Product and try to see its proof, they putenter image description here and I don't see any proof for $\det(EA) = \det(E)\det(A)$, so it circles back to the question - Proof $\det(AB)=\det(A)\det(B)$. I haven't learned about permutations, so I have trouble understanding the proof 1 section of the wiki link. I would be glad if you show me a proof for $\det(EA)=\det(E)\det(A)$ without permutations.

My previous question was Is there any easy way to prove $\det(EA) = \det(E)\det(A)?$, where $E$ is an elementary matrix?. But I found that it was not a duplicated question-please check the comments-so I am asking the question again.

I checked Proof of determinants for matrices of any order, Proof $\det(AB)=\det(A)\det(B)$, and How to show that $\det(AB) =\det(A) \det(B)$?, but they don't explain what I want to know.

user3315463
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  • I've seen this proof done by cases. There are only so many different types of elementary matrices, and there are nice formulas for the entries of $ED$ in terms of the entries of $D$. I think the proof works out most nicely when using the sum over permutations definition of the determinant. – TomKern May 13 '21 at 02:19
  • If you haven't learned about permutations, what definition of the determinant are you using? – Hans Lundmark May 13 '21 at 05:35
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    @HansLundmark I asked an almost identical question last time momo asked this, so I don't know why that wasn't in the body of the question this time around... In any case, OP said they defined the determinant through cofactor expansion. – Brian Moehring May 13 '21 at 06:00

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