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Question for a theorem proof

  1. Aug 6, 2009 #1
    1. The problem statement, all variables and given/known data
    Let pA(x) and pB(x) be the characteristic polynomial for A and B. If B is similar to A, then there exists a non-singular matrix S so B = S-1AS. Thus there:

    pB(x) = det(B-[tex]\lambda[/tex]I)
    = det(S-1AS-[tex]\lambda[/tex]I)
    = det(S-1(A-[tex]\lambda[/tex]I)S)
    = det(S-1)det(A-[tex]\lambda[/tex]I)det(S)
    = pA(x)

    2. Relevant equations

    3. The attempt at a solution

    My question is, that I really don't know why the 3rd step is legal ? Can I just put the S's anywhere I want ?

  2. jcsd
  3. Aug 6, 2009 #2


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    Science Advisor
    Homework Helper

    No, you can't 'put S's anywhere you want'. That would be silly. What you can do is use that I=S^(-1)*I*S and use the rules of algebra to move the S's around by factoring.
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