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Math Help - Matrix Multiplication and Solving for X?

  1. #1
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    Matrix Multiplication and Solving for X?

    Find X if

    [5 -8]X + [0 9] = [-3 6]
    [8 -1].... [0 -8] ....[-1 -9]

    X =?

    My steps:

    [5 -8]X + [0 9] = [-3 6]
    [8 -1] ......[0 -8] ..[-1 -9]

    Ax + B = C

    Subtract matrix B on both sides gives

    [5 -8]X = [-3 6] - [0 9]
    [8 -1] .......[-1 -9] [0 -8]

    [5 -8]X = [-3 -3]
    [8 -1] .......[-1 -1]

    A^-1 = 1/(ad -bc) [d -b]
    [-c a]

    [5 -8] = 1/(5*-1 - -8*8) [-1 8]
    [8 -1] .............................[-8 5]

    =1/59[-1 8]
    ..........[-8 5]

    X = [-3 -3] * 1/59[-1 8]
    .....[-1 -1] .............[-8 5]

    = 1/59 [(-3*-1 + -3*-8) (-3*8 + -3*5)]
    ............[(-1*-1 + -1*-8) (-1*8 + -1*5)]

    X= 1/59 [27 -39]
    ...............[9 - 13]

    But for some reason that is not the answer for X, can someone help me please? Thank you!
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  2. #2
    MHF Contributor

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    Re: Matrix Multiplication and Solving for X?

    here is what you did wrong:

    AX + B = C

    AX = C - B <---you're fine up to here

    A-1(AX) = A-1(C - B) <--you multiplied C - B on the wrong side.

    X = A-1(C - B).

    if you multiply C - B on the LEFT by A-1, you get:

    X = \frac{1}{59} \begin{bmatrix}-1&8\\-8&5 \end{bmatrix} \begin{bmatrix}-3&-3\\1&-1 \end{bmatrix} = \frac{1}{59}\begin{bmatrix} 11&-5\\29&19 \end{bmatrix}

    with matrices, AB is NOT the same as BA.
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