# Homework Help: Similar matrices

1. Mar 20, 2010

### Dustinsfl

Show that if A and B are similar matrices, then the det(A)=det(b).

I am not entirely sure how to start this proof.
I was thinking...
A=ai j

det(A)=$$\sum$$ai j*(-1)i+j*Mi j from j=1 to n.

I am pretty much clueless on this one though and not sure if I am just throwing stuff on the wall hoping something will stick

2. Mar 20, 2010

### marcusl

Write down the definition of similar matrices. What do you know about determinants that you can apply to that equation?

3. Mar 20, 2010

### Dustinsfl

Matrix A is similar to B if there exists S such that B=S-1*A*S

4. Mar 20, 2010

### Dustinsfl

det(B)=det(S-1*A*S)=det(S-1)det(A)det(S)=det(S-1*S)det(A)=det(I)det(A)=det(A) thus, det(B)=det(A)

Remarkable easy.

5. Mar 20, 2010

Good job!