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elementary row operations

When we perform these operations:
Ri+αRjR_i + \alpha R_j Determinant does not change
αRi\alpha R_i Determinant multiplies by α\alpha
RiRjR_i \Leftrightarrow R_j Determinant multiplies by (-1)


I know these are true, however, I'm looking for proofs for each of them. They aren't anywhere in my lecture notes and I can't find anything online, all I can find is what happens to the determinant, not the actual proof.

Has anyone got anything that could help? Thank you :smile:
Reply 1
Original post by hey_venus
When we perform these operations:
Ri+αRjR_i + \alpha R_j Determinant does not change
αRi\alpha R_i Determinant multiplies by α\alpha
RiRjR_i \Leftrightarrow R_j Determinant multiplies by (-1)


I know these are true, however, I'm looking for proofs for each of them. They aren't anywhere in my lecture notes and I can't find anything online, all I can find is what happens to the determinant, not the actual proof.

Has anyone got anything that could help? Thank you :smile:


If you know what an elementary matrix is, you could look at their determinants for the different operations. As doing a row operation equals multiplying your matrix AA with your relevant elementary matrix EE, you get the new matrix AEAE, and det(AE)=det(A)det(E)det(AE) = det(A)det(E). However, the proofs for these are rather long (at least the ones I've come across), so it would take forever to prove them in detail (but this is the whole concept, the proofs aren't hard to understand, so I think you could get it by looking at their determinants).

Hopefully this helped, and good luck :smile:

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