Prove that the determinants of similar matrices are equal

[BraedenP]

Homework Statement

I'm supposed to write a proof for the fact that det(A)=det(B) if A and B are similar matrices.

Homework Equations

Similar matrices have an invertible matrix P which satisfies the following formula:
A=PBP^{-1}

det(AB) = det(A)det(B)

The Attempt at a Solution

Basically, I rearranged the above formulae to do the following:

A=PBP^{-1}

AP=PB

det(AP)=det(PB)

det(A)det(P)=det(P)det(B)

At this point, everything is scalar, so the det(P) on each side cancel, leaving det(A)=det(B)

My question is.. Is this sufficient proof, or is more required?

 

[HallsofIvy]

Yes, just add the point that since P is invertible, its determinant is non-zero and so you can divide both sides of the equation by det(P).

In fact, it might be simpler to not change to "AP= PB" at all.

From A= PBP^{-1}, you have det(A)= det(P)det(B)det(P^{-1}). Now, you have det(P^{-1}= 1/det(P) and, since those are numbers, multiplication is commutative.

 

[BraedenP]

Yeah, true. Thanks. :)