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Math Help - Using linear equations to solve unknowns in matrices

  1. #1
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    Using linear equations to solve unknowns in matrices

    Question 87

    Find the solution of the following set of linear equations.

    [ 1 2 1 4 ] {x1} = { 13 }
    [ 2 0 4 3 ] {x2} = {28.3}
    [ 4 2 2 1 ] {x3} = {20.9}
    [ -3 1 3 2 ] {x4} = {6.5 }

    Just to be clear there are three matrices, of the orders 4x4, 4x1 and 4x1. I've managed to split them into the linear equations, but just looking at them make me sweat with fear.

    x1 + 2 x2 + x3 + 4 x4 = 13
    2 x1 + 4 x3 + 3 x4 = 28.3
    4 x1 + 2 x2 + 2 x3 + x4 = 20.9
    -3 x1 + x2 + 3 x3 + 2 x4 = 6.5
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  2. #2
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    Hello, Trapper Dave!

    You didn't explain what method you're supposed to use . . .


    Question 87

    Find the solution of the following set of linear equations:

    . . \begin{array}{cccc}w + 2x + y + 4z & = & 13 \\ 2w \qquad + 4y + 3z & = & 28.3 \\ 4w + 2x + 2y + z & = & 20.9 \\ \text{-}3w + x + 3y + 2z & = & 6.5\end{array}

    \begin{pmatrix}1&2&1&4\\2 &0&4&3\\4&2&2&1\\\text{-}3&1&3&2\end{pmatrix}\begin{pmatrix}w\\x\\y\\z\end  {pmatrix} \;=\;\begin{pmatrix}13 \\ 28.3 \\ 20.9 \\ 6.5\end{pmatrix}

    We can solve the system using "normal" algebra . . . lots of it!


    If we want a matrix solution, we need the inverse of the 4-by-4 matrix.

    We'd work with: . \begin{vmatrix}1&2&1&4&|&1&0 &0&0\: \\ 2&0&4&3&|&0&1&0&0\: \\ 4&2&2&1&|&0&0&1&0\: \\ \text{-}3&1&3&2&|&0&0&0&1\:\end{vmatrix}

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  3. #3
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    So that would give me;



    What should I do next? Any method at all will do.
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  4. #4
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    Quote Originally Posted by Trapper Dave View Post
    What should I do next? Any method at all will do.
    CV = D\quad  \Rightarrow \quad V = C^{ - 1} D
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  5. #5
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    Quote Originally Posted by Trapper Dave View Post
    So that would give me;



    What should I do next? Any method at all will do.
    If you did the inversion using the Gauss-Jordan method you would have
    saved some work by doing it with:

    .  \left( \begin{array}{cccc|c}1&2&1&4&13 \\ 2&0&4&3&28.3 \\ 4&2&2&1&20.9 \\ \text{-}3&1&3&2&6.5\end{array} \right)

    as this will solve the equations directly without the need for a further matrix
    multiplication.

    RonL
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