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Linear Algebra - finding matrix A

  1. Sep 17, 2012 #1
    1. The problem statement, all variables and given/known data
    Find A if (2A-1 - 3I)T =

    2*[tex]
    \begin{pmatrix}
    -1 & 2\\
    5 & 4
    \end{pmatrix}
    [/tex]

    2. Relevant equations



    3. The attempt at a solution

    I have no idea if I'm even on the right track of solving this question...

    I simplified the right hand side down to
    [tex]
    \begin{pmatrix}
    -2 & 4\\
    10 & 8
    \end{pmatrix}
    [/tex]

    I took the transpose of both sides so the left hand side is just (2A-1 - 3I)
    The right hand side is now
    [tex]
    \begin{pmatrix}
    -2& 10\\
    4 & 8
    \end{pmatrix}
    [/tex]

    I multiplied both sides by A so
    LH: A(2A-1 - 3I)
    RH: A*[tex]
    \begin{pmatrix}
    -2& 10\\
    4 & 8
    \end{pmatrix}
    [/tex]

    Distribute the A so
    LH: (A2A-1 - A3I) => (2I - A3I)

    Bring the A3I part to the right side
    LH = 2I
    RH = A3I + A*[tex]
    \begin{pmatrix}
    -2& 10\\
    4 & 8
    \end{pmatrix}
    [/tex]

    Simplify the A3I part on the right hand side to get
    A * [tex]
    \begin{pmatrix}
    3 & 0\\
    0 & 3
    \end{pmatrix}
    [/tex]

    Factor out the A on the right hand side to get
    A * ([tex]
    \begin{pmatrix}
    -2 & 10\\
    4 & 8
    \end{pmatrix}
    [/tex]

    +
    [tex]
    \begin{pmatrix}
    3 & 0\\
    0 & 3
    \end{pmatrix}
    [/tex]
    )

    Add the two matrices together in the brackets to get
    A * [tex]\begin{pmatrix}
    1 & 10\\
    4 & 11
    \end{pmatrix}
    [/tex]

    So I'm left with
    [tex]
    \begin{pmatrix}
    2 & 0\\
    0 & 2
    \end{pmatrix}
    [/tex]

    =

    A *
    [tex]
    \begin{pmatrix}
    1 & 10\\
    4 & 11
    \end{pmatrix}
    [/tex]

    Now there's no way I can somehow get A like that so I screwed up somewhere in there.. probably from the start lol

    This looks so messy... am I allowed to upload my handwritten work on a site then post it here? Seems like a better alternative to the mess I have up there >_> Anyways I'm not really looking for a step by step on how to do this. Just a push in the right direction is all...
     
  2. jcsd
  3. Sep 17, 2012 #2

    Mark44

    Staff: Mentor

    It looks like you did a lot more work than you needed to.

    You have
    $$ 2A^{-1} - 3I = \begin{pmatrix}
    -2& 10\\
    4 & 8
    \end{pmatrix}$$

    Just add 3I to both sides, and then you'll have an expression for 2A-1, from which you can fairly easily find A.
     
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