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. :)