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Math Help - Matrix help

  1. #1
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    Matrix help

    Hi guys. Strange question here but i dont know how to solve and its for revision.

    let A = 2x2 matrix ( 2 is the top left entry, 1 the top right entry, 7 the bottom left and 4 the bottom right)

    B = 2x2 matrix (3 the top left entry, x the top right, y the bottom left and 4 the bottom right)

    determine all the possible values for x and y, but such that AB = BA


    please help this is a revision question for an upcoming exam so i really need to know how to do it. thanks!
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  2. #2
    Member sbhatnagar's Avatar
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    Re: Matrix help

    A=\begin{bmatrix}2 & 1 \\  7 & 4\end{bmatrix}

    B=\begin{bmatrix}3 & x \\  y & 4\end{bmatrix}

    AB=\begin{bmatrix}2 & 1 \\  7 & 4\end{bmatrix} \begin{bmatrix}3 & x \\  y & 4\end{bmatrix}=\begin{bmatrix} 6+y & 2x+14 \\ 21 + 4y & 7x+16\end{bmatrix}

    BA=\begin{bmatrix}3 & x \\  y & 4\end{bmatrix}\begin{bmatrix}2 & 1 \\  7 & 4\end{bmatrix} =\begin{bmatrix} 6+7x & 3+4x \\ 2y+28 & y+16\end{bmatrix}

    According to the question,

    \begin{bmatrix} 6+y & 2x+14 \\ 21 + 4y & 7x+16\end{bmatrix}=\begin{bmatrix} 6+7x & 3+4x \\ 2y+28 & y+16\end{bmatrix}

    Therefore you have the following equations,

    6+y=6+7x \\ 2x+14=3+4x \\ 21+4y=2y+28 \\ 7x+16=y+16
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  3. #3
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    Re: Matrix help

    But how do i get x and y from there?

    I tried one way but i dont know how to find the inverse of a four square (and im not meant to in my course)
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  4. #4
    Member Goku's Avatar
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    Re: Matrix help

    Quote Originally Posted by sbhatnagar View Post
    A=\begin{bmatrix}2 & 1 \\  7 & 4\end{bmatrix}

    B=\begin{bmatrix}3 & x \\  y & 4\end{bmatrix}

    AB=\begin{bmatrix}2 & 1 \\  7 & 4\end{bmatrix} \begin{bmatrix}3 & x \\  y & 4\end{bmatrix}=\begin{bmatrix} 6+y & 2x+14 \\ 21 + 4y & 7x+16\end{bmatrix}
    6+y=6+7x \\ 2x+14=3+4x \\ 21+4y=2y+28 \\ 7x+16=y+16


    AB=\begin{bmatrix}2 & 1 \\  7 & 4\end{bmatrix} \begin{bmatrix}3 & x \\  y & 4\end{bmatrix}=\begin{bmatrix} 6+y & 2x+4 \\ 21 + 4y & 7x+16\end{bmatrix}

    6+y=6+7x \\ 2x+4=3+4x \\ 21+4y=2y+28 \\ 7x+16=y+16
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  5. #5
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    Re: Matrix help

    you're not meant to find the inverse of anything.

    from 6 + y = 6 + 7x, we have:

    y = 7x. so if we find out x, we will know y.

    from 2x + 4 = 3 + 4x we get:

    2x - 1 = 0, so x = 1/2.

    we need to check the last two equations for consistency:

    21 + 4(7/2) = 21 + 14 = 35
    2(7/2) + 28 = 7 + 28 = 35, the 3rd equation is consistent with the first 2.

    7(1/2) + 16 = 7/2 + 32/2 = 39/2
    7/2 + 16 = 7/2 + 32/2 = 39/9, all 4 equations are consistent.

    therefore B can only be:

    [3 1/2]
    [7/2 4]

    (no inverses were harmed in the making of this film)
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