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Inverse of matrix AB?

  1. Apr 11, 2009 #1
    1. The problem statement, all variables and given/known data

    Use the given matrices to find (AB)^-1

    (These are 2 X 2 matrices, please ignore the fraction bar in between the top and bottom elements. I can't figure this stupid latex piece of crap out)

    A[tex]^{-1}[/tex] = [tex]\left(\frac{\frac{1}{2}}{\frac{-1}{2}} \ldots \frac{\frac{-5}{2}}{\frac{3}{2}}\right)[/tex]

    B[tex]^{-1}[/tex] = [tex]\left(\frac{\frac{2}{3}}{\frac{-1}{3}} \ldots \frac{\frac{4}{3}}{\frac{5}{2}}\right)[/tex]

    2. Relevant equations

    Only the one for inverse matrices which states,

    (AB)[tex]^{-1}[/tex] = B[tex]^{-1}[/tex]A[tex]^{-1}[/tex]

    3. The attempt at a solution

    The answer I get in the end is:

    (AB)[tex]^{-1}[/tex] = [tex]\left(\frac{\frac{-1}{3}}{\frac{-17}{12}} \ldots \frac{\frac{1}{3}}{\frac{55}{12}}\right)[/tex]

    But the book gets,

    (AB)[tex]^{-1}[/tex] = [tex]\left(\frac{\frac{-1}{3}}{-1} \ldots \frac{\frac{1}{3}}{\frac{10}{3}}\right)[/tex]

    Am I the one doing something wrong, or is the book wrong?

    Any help would be greatly appreciated.

    Thanks!
     
  2. jcsd
  3. Apr 11, 2009 #2

    CompuChip

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    Homework Helper

    If you copied the matrices correctly, your answer is correct.
    If the 5/2 in (2,2)-entry of [itex]B^{-1}[/itex] is supposed to be 5/3 then the book is correct.

    By the way, click the formula to see the LaTeX code:
    [tex]
    A^{-1} = \begin{pmatrix} \frac{1}{2} & -\frac52 \\ \-\frac{1}{2} & \frac{3}{2} \end{pmatrix} =
    \frac12 \begin{pmatrix}
    1 & -5 \\
    -1 & 3
    \end{pmatrix}
    [/tex]
     
  4. Apr 11, 2009 #3
    Nope, the book says what I wrote. I'm not surprised though; I've found several typos in the questions in this particular textbook...

    Thanks a lot for the clarification, I thought maybe I was missing some obscure rule :p.

    Also, thanks for showing how to use the LaTex code properly :)
     
  5. Apr 26, 2009 #4
    I find it a lot more easy to calculate AB first,then the inverse...

    just a tip,since the main question was already answered
     
  6. Apr 30, 2009 #5

    CompuChip

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    In this case your approach would take longer, because you first have to calculate A and B from their inverses, then multiply them and finally take the inverse of that, while just multiplying the two given matrices in the correct order gives you the right answer immediately.
     
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