Can the Determinant of Similar Matrices be Proven Equal?

[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=a_{i j}

det(A)=\suma_{i j}*(-1)^i+j*M_{i 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

 

[marcusl]

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

 

[Dustinsfl]

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

 

[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.

 

[marcusl]

Good job!