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Matrix Equation AXB=C solve for X

  1. Jun 29, 2012 #1
    Hi

    I have a question in my math coursework on matrices
    Question is
    Three matrices A, B and C are given by:

    [tex]A = \begin{pmatrix} 2 & 5 \\ -4 & 1 \end{pmatrix} [/tex]
    [tex]B = \begin{pmatrix} 1 & 3 \\ -2 & 6 \end{pmatrix}[/tex]
    [tex]C = \begin{pmatrix} 2 & 1 \\ -3 & -1 \end{pmatrix}[/tex]

    a) find the inverses of A and B
    b) Solve the equation AXB=C

    This is my solution

    iknow have rearranged equation to make it

    [itex]{X=}A\frac{C}{A B} [/itex]

    i understand that you can't dived in matrices so i have to use the inverse of A and B
    to make the equation [itex]{X=}{A}^{-1} {C} {B}^{-1} [/itex]

    The det [itex]\left|A\right|[/itex] = 22
    det [itex]\left|B\right|[/itex] = 12

    [itex]A^{-1} = \frac{1}{22}[/itex] \begin{pmatrix} 1 & -5 \\ 4 & 2 \end{pmatrix}
    [itex]B^{-1} = \frac{1}{12} [/itex] /begin{pmatrix} 6 & -3 \\ 2 & 1 \end{pmatrix}

    and my answer for inv A mutliplied C is \frac{1}{22}[/itex] \begin{pmatrix} 17 & 6 \\ 2 & 2 \end{pmatrix} or \begin{pmatrix} 0.7727 & 0.2727 \\ 0.0909 & 0.0909\end{pmatrix}

    and then i mutlipy by inv B i get matrix X= \begin{pmatrix} 0.4318 & -0.1705 \\ 0.0606 & -0.0152\end{pmatrix}

    my question is this answer correct and how can i show the answer for Matrix X as [itex]X=\frac{1}{?} [/itex]\begin{pmatrix} a & b \\ c & d\end{pmatrix}

    do i add the [itex]\frac {1}{22}+ \frac{1}{12} [/itex]which would make it [itex] \frac{1}{34} ??????[/itex]

    would this then make Matrix X [itex]X=\frac{1}{34} [/itex]\begin{pmatrix} a & b \\ c & d\end{pmatrix}
     
    Last edited by a moderator: Jun 29, 2012
  2. jcsd
  3. Jun 29, 2012 #2

    tiny-tim

    User Avatar
    Science Advisor
    Homework Helper

    Welcome to PF!

    Hi Sage Sky! Welcome to PF! :smile:

    If you're having trouble writing matrices in latex,

    then try the CODE button instead, like this …

    Code (Text):

    1 2 3    1 3 2
    4 5 6    4 6 7
    7 8 9    9 8 7
    :wink:
     
  4. Jun 29, 2012 #3
    thx got it except its goes to new line on matrixs
     
  5. Jun 29, 2012 #4

    tiny-tim

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    Science Advisor
    Homework Helper

    Hi Sage Sky! :smile:
    (i haven't checked your figures, but the method looks ok)

    you could have kept the factors 1/22 and 1/12 outside the matrices all the time :wink:

    (so you'd be multiplying them in the end)
     
  6. Jun 29, 2012 #5

    Borek

    User Avatar

    Staff: Mentor

    It doesn't. You were doing strange tricks to the opening and closing tags (you were opening them and closing them in random places).

    And be consistent - don't mix LaTeX with non LaTeX formulas.
     
  7. Jun 29, 2012 #6
    ok so it would be [itex] \frac{1}{264} \begin{pmatrix} 114 & -45 \\ 16 & -4\end{pmatrix}[/itex]
     
  8. Jun 29, 2012 #7

    Mark44

    Staff: Mentor

    You can check your answer by multiplying AXB and comparing it to C. Don't bring the fractions into your matrices - you can move them all the way to the left and multiply all of them together and all the matrices together.
     
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