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Solving for B matrix in terms of A. Can someone check my answer?

  1. Sep 15, 2012 #1
    1. The problem statement, all variables and given/known data

    Suppose P is invertible and A = PBP^-1. Solve for B in terms of A.

    2. Relevant equations

    (AB)^-1 = B^-1*A^-1

    3. The attempt at a solution

    multiply from the left of each side of the equation by P^-1:
    P^-1 *A = BP^-1

    Take the inverse of both sides:

    (P^-1*A)^-1 = (BP^-1)^-1
    A^-1 * P = P*B^-1

    Multiply from the left of each side of the equation by P^-1:
    P^-1*A^-1*P = B^-1

    Take the inverse of both sides:

    [P^-1(A^-1*P)]^-1 = B

    B = (A^-1*P)^-1*P
    B = P^-1 *A * P


    Is that correct?
     
  2. jcsd
  3. Sep 15, 2012 #2

    LCKurtz

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    Yes, it looks OK but is much more complicated than it needs to be. At the first step where you have ##P^{-1}A= BP^{-1}## just multiply by ##P## on the right.
     
  4. Sep 15, 2012 #3
    Hahaaaa. That's right. Thanks. :]
     
  5. Sep 15, 2012 #4
    I wasn't sure if you can multiply from the right too.
     
  6. Sep 15, 2012 #5

    Ray Vickson

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    Why not? Two matrices can be multiplied as long as their row and column numbers match up properly. If all three of A, B and P (and P^(-1)) are nxn they can be multiplied in any order.

    RGV
     
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