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Homework Help: Linear Algebra and Canceling

  1. Sep 13, 2010 #1
    1. The problem statement, all variables and given/known data

    From AB=C find a formula for A^-1. Also, from PA=LU find A^-1

    3. The attempt at a solution

    My answers were (A^-1)=(C^-1)(B) and (L^-1)(U^-1)(P)=(A^-1).

    The answers in the back of the book are (A^-1)=(B)(C^-1) and (A^-1)=(U^-1)(L^-1)(P)

    Would I say that my canceling of matrices is wrong? Would I call what I am trying to do 'cancelling matrices'? What should I be considering when trying to carry out operations such as these?
     
  2. jcsd
  3. Sep 14, 2010 #2

    Office_Shredder

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    Why don't you show what steps you did to get those answers?
     
  4. Sep 14, 2010 #3
    Good point.

    First: AB = C --> (A^-1)(A)(B) = (A^-1)(C) = (B) --> (A^-1)=(C^-1)(B)

    Second: PA = LU --> [want to cancel A on left side so I multiplied by A^-1 on both sides. assumed placement of A^-1 on left side didn't matter, as long as I could cancel] (P)(A^-1)(A)=(A^-1)(L)(U) --> [basically did the same thing as in the previous step, multiplying first by the inverse of the matrix closest to A^-1, which was L] (L^-1)(U^-1)(P)=(A^-1)

    Let me know if I can make things clearer.
     
  5. Sep 14, 2010 #4

    Mark44

    Staff: Mentor

    The placement of a matrix in matrix multiplication does matter, since matrix multiplication is not generally commutative.

    For the first problem (and maybe you can get an idea for the second),

    AB = C
    ==> A = CB-1
    ==> A-1 = (CB-1)-1 = (B-1)-1C-1
    ==> A-1 = BC-1

    In the second step I multiplied on the right by B-1. In the third step I used the theorem that (AB)-1 = B-1C-1.
     
  6. Sep 14, 2010 #5
    Okay. So...

    PA=LU

    A=P-1LU

    A-1=(P-1LU)-1

    A-1=U-1L-1P

    ...?

    It seems like whatever placement I use to cancel something out (to the left or right of the term) I use that same placement (to the left or right of all the factors) on the other side of the equation.
     
  7. Sep 14, 2010 #6

    Office_Shredder

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    That makes sense. If you take your term on the left, and your term on the right, and call them M and N for a second

    We have that M=N

    So if we want to cancel say a matrix C, we need to multiply one of these by C-1. MC-1=NC-1 just by directly replacing M with N. So the upshot is that whenever you multiply a matrix equation by another matrix, you have to make sure what you're multiplying by is either at the start or at the end of both expressions
     
  8. Sep 14, 2010 #7

    Mark44

    Staff: Mentor

    Which agrees with the answer in the book.
     
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