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Similar matrices

  • Thread starter Dustinsfl
  • Start date
  • #1
699
5
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)=[tex]\sum[/tex]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
 

Answers and Replies

  • #2
marcusl
Science Advisor
Gold Member
2,714
382
Write down the definition of similar matrices. What do you know about determinants that you can apply to that equation?
 
  • #3
699
5
Matrix A is similar to B if there exists S such that B=S-1*A*S
 
  • #4
699
5
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
marcusl
Science Advisor
Gold Member
2,714
382
Good job!
 

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