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Math Help - I can't solve this matrix using the Gauss-Jordan method, i need help!!

  1. #1
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    I can't solve this matrix using the Gauss-Jordan method, i need help!!

    { 2 -2 -4 | -2
    -2 3 1 | 7
    3 -3 -6 | -3}



    I CANT FIGURE THIS ONE OUT! I can only do it on a calculator, obviously without showing work, but, i need to figure this problem out showing all steps. I would really appreciate it if someone would be able to help me out, thanks.
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  2. #2
    Member kjchauhan's Avatar
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    matrix

    The given matrix is a singular matrix, so solution does not exist.
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  3. #3
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    Hello, burdonjo!

    This system has an infinite number of solutions.


    Solve: . \left|\begin{array}{ccc|c}2 & \text{-}2& \text{-}4 & \text{-}2 \\ \text{-}2 & 3 & 1 & 7 \\ 3 & \text{-}3 & \text{-}6 & \text{-}3\end{array}\right|

    . . . \begin{array}{c}\frac{1}{2}R_1 \\ \\ \frac{1}{3}R_3\end{array} \left|\begin{array}{ccc|c}1 & \text{-}1 & \text{-}2 & \text{-}1 \\ \text{-}2 & 3 & 1 & 7 \\ 1 & \text{-}1 & \text{-}2 & \text{-}1 \end{array}\right|

    \begin{array}{c}\\ R_2 + 2R_1 \\ R_3 - R_1\end{array} \left|\begin{array}{ccc|c}1 & \text{-}1 & \text{-}2 & \text{-}1 \\ 0 & 1 & \text{-}3 & 5 \\ 0 & 0 & 0 & 0 \end{array}\right|

    . \begin{array}{c}R_1+R_2 \\ \\ \\ \end{array} \left|\begin{array}{ccc|c}1 & 0 & \text{-}5 & 4 \\ 0 & 1 & \text{-}3 & 5 \\ 0 & 0 & 0 & 0 \end{array}\right|


    We have:. . \begin{Bmatrix}x - 5z &=& 4 \\ y -3x &=& 5 \end{Bmatrix}\quad\Rightarrow\quad \begin{Bmatrix}x &=& 5z + 4 \\ y &=& 3z + 5 \\ z &=& z \end{Bmatrix}


    On the right, replace z with a parameter t.

    . . . . \begin{Bmatrix}x &=& 5t + 4 \\ y &=& 3t + 5 \\ z &=& t \end{Bmatrix}

    This represents all the solutions for the system of equations,
    . . one solution for each value of t.

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  4. #4
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    Bottom row

    When the bottom row of the matrix is zeros all the way across, does that mean that the matrix has "no solution"? Sorry to bother you
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  5. #5
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    Quote Originally Posted by burdonjo View Post
    When the bottom row of the matrix is zeros all the way across, does that mean that the matrix has "no solution"? Sorry to bother you
    Are you talking about the row reduced coefficient matrix or the row reduced augmented matrix (with the right side of the system of equations as the last column)? If any column of the row reduced augmented matrix has all "0"s except that the last column is not 0, then there exist no solution. If all rwos of the row reduced augmented matrix that have all "0" also have "0" in the last column, then there exist an infinite number of solutions.

    In either case, there is not a "unique" solution.
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