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Similar matrices

  1. Mar 20, 2010 #1
    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
     
  2. jcsd
  3. Mar 20, 2010 #2

    marcusl

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    Write down the definition of similar matrices. What do you know about determinants that you can apply to that equation?
     
  4. Mar 20, 2010 #3
    Matrix A is similar to B if there exists S such that B=S-1*A*S
     
  5. Mar 20, 2010 #4
    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.
     
  6. Mar 20, 2010 #5

    marcusl

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    Good job!
     
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