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Math Help - Simultaneous equations to matrix

  1. #1
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    Simultaneous equations to matrix

    So I'm given this question.

    4x + 2y + 4x = 3, 2y + z = 1, 2x + 2z = 1

    and it says Forumlate a single vector matrix equation from the three equations given.

    Sounds pretty obvious but a single vector matrix would just be

    |x| = |3|
    |y| = |1|
    |z| = |1|

    Sorry I can't figure out how to set the proper array brackets and stuff. But that is the single matrix vector?

    Here is a correct inverse matrix of the matrix you formulated obove.

    1 -1 -1.5
    0.5 0 -1
    -1 1 2

    Sorry about the formatting lol. I think I'm missing a step though because I have to show that this matrix is a correct inverse of the matrix I created obove, but the matrix I created was just a single vector??
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  2. #2
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    Re: Simultaneous equations to matrix

    I don't think this intended a matrix made of a single vector. Rather, they want a single equation (which happens to be a "vector-matrix" equation) rather than three equations.
    And that would be
    \begin{bmatrix}4 & 2 & 4 \\ 0 & 2 & 1\\ 2 & 0 & 2\end{bmatrix}\begin{bmatrix}x \\ y \\ z\end{bmatrix}= \begin{bmatrix}3 \\ 1 \\ 1\end{bmatrix}.
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    Re: Simultaneous equations to matrix

    Quote Originally Posted by uperkurk View Post
    So I'm given this question.

    4x + 2y + 4z = 3, 2y + z = 1, 2x + 2z = 1
    M = \left( {\begin{array}{*{20}{c}} 4&2&4\\ 0&2&1\\ 2&0&2\end{array}} \right)

    You want to find {M^{ - 1}}\left( {\begin{array}{*{20}{c}} 3\\ 1\\ 1 \end{array}} \right)
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    Re: Simultaneous equations to matrix

    Quote Originally Posted by HallsofIvy View Post
    I don't think this intended a matrix made of a single vector. Rather, they want a single equation (which happens to be a "vector-matrix" equation) rather than three equations.
    And that would be
    \begin{bmatrix}4 & 2 & 4 \\ 0 & 2 & 1\\ 2 & 0 & 2\end{bmatrix}\begin{bmatrix}x \\ y \\ z\end{bmatrix}= \begin{bmatrix}3 \\ 1 \\ 1\end{bmatrix}.

    oh right yeh that makes a lot more sense now. Thanks.

    @Plato, I can verify that the second part of the question is true. It says show that the following matrix is a correct inverse of the matrix obove.

    \begin{bmatrix}1 & -1 & -1.5 \\ 0.5 & 0 & -1\\ -1 & 1 & 2\end{bmatrix} by using an indentiy matrix \begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0\\ 0 & 0 & 1\end{bmatrix} so multiplying that matrix by the identity matrix should return \begin{bmatrix}3 \\ 1 \\ 1\end{bmatrix}. ?
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    Re: Simultaneous equations to matrix

    Quote Originally Posted by uperkurk View Post
    @Plato, I can verify that the second part of the question is true. It says show that the following matrix is a correct inverse of the matrix obove.
    \begin{bmatrix}1 & -1 & -1.5 \\ 0.5 & 0 & -1\\ -1 & 1 & 2\end{bmatrix} by using an indentiy matrix \begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0\\ 0 & 0 & 1\end{bmatrix} so multiplying that matrix by the identity matrix should return \begin{bmatrix}3 \\ 1 \\ 1\end{bmatrix}. ?
    Show that if M^{-1}=\begin{bmatrix}1 & -1 & -1.5 \\ 0.5 & 0 & -1\\ -1 & 1 & 2\end{bmatrix} then
    M\cdot M^{-1}=M^{-1}\cdot M=\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0\\ 0 & 0 & 1\end{bmatrix}
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    Re: Simultaneous equations to matrix

    What the freak! I know how to calculate a matrix but for some reason when I'm doing it manually

    \begin{bmatrix}4 & 2 & 4 \\ 0 & 2 & 1\\ 2 & 0 & 2\end{bmatrix} X \begin{bmatrix}1 & -1 & -1.5 \\ 0.5 & 0 & -1\\ -1 & 1 & 2\end{bmatrix} but I end up with 8 for the first position wtf...?

    4 x 1 + 2 x 0.5 + 4 x -1 = 8? but it should equal 1?

    when I use the online calculator it comes out as \begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0\\ 0 & 0 & 1\end{bmatrix}

    So what is going on?
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    Re: Simultaneous equations to matrix

    Quote Originally Posted by uperkurk View Post
    What the freak! I know how to calculate a matrix but for some reason when I'm doing it manually

    \begin{bmatrix}4 & 2 & 4 \\ 0 & 2 & 1\\ 2 & 0 & 2\end{bmatrix} X \begin{bmatrix}1 & -1 & -1.5 \\ 0.5 & 0 & -1\\ -1 & 1 & 2\end{bmatrix} but I end up with 8 for the first position wtf...?

    4 x 1 + 2 x 0.5 + 4 x -1 = 8? but it should equal 1?

    when I use the online calculator it comes out as \begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0\\ 0 & 0 & 1\end{bmatrix}

    So what is going on?
    Arithmetic (4)( 1) + (2)(0.5) + (4)(-1) = (4)+(1)+(-4)=1
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  8. #8
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    Re: Simultaneous equations to matrix

    omg I feel utterly retarded now -_-
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