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Math Help - Homogenous system of linear equations.

  1. #1
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    Homogenous system of linear equations.

    So I've been doing this for a VERY long time now I don't seem to be getting the right answer (according to online solvers) I'm wondering if you guys can spot my error because this is REALLY frustrating.
    A=\left(\begin{matrix} 2/5&1/5\\3 &0\end{matrix}\right)
    And I found one of the Eigenvalues to be 1. so now subtracting the Diagonal from 1 I get this and I start to solve the homogeneous system of equation.
    (L=1)=\left(\begin{matrix} 3/5&1/5\\3 &1\end{matrix}\right) = \left(\begin{matrix} 3&1\\3 &1\end{matrix}\right) = \left(\begin{matrix} 1&1/3\\0 &0\end{matrix}\right)
    Now from here x(2)=T and x(1)+(1/3)T=0 --->X(1)=(-1/3)T
    SO X=\left(\begin{matrix} -1\\3\end{matrix}\right) BUT according toe the online solver it should be X=\left(\begin{matrix} 1\\3\end{matrix}\right)
    I have tried everything so I need someone's help, WHAT AM I DOING WRONG?
    Thank you!
    Last edited by mr fantastic; October 1st 2011 at 01:52 PM. Reason: Re-titled.
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  2. #2
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    Re: Homogenous system of linear equations.

    Quote Originally Posted by calculuskid1 View Post
    So I've been doing this for a VERY long time now I don't seem to be getting the right answer (according to online solvers) I'm wondering if you guys can spot my error because this is REALLY frustrating.
    A=\left(\begin{matrix} 2/5&1/5\\3 &0\end{matrix}\right)
    And I found one of the Eigenvalues to be 1. so now subtracting the Diagonal from 1 I get this and I start to solve the homogeneous system of equation.
    (L=1)=\left(\begin{matrix} 3/5&1/5\\3 &1\end{matrix}\right) = \left(\begin{matrix} 3&1\\3 &1\end{matrix}\right) = \left(\begin{matrix} 1&1/3\\0 &0\end{matrix}\right)
    Now from here x(2)=T and x(1)+(1/3)T=0 --->X(1)=(-1/3)T
    SO X=\left(\begin{matrix} -1\\3\end{matrix}\right) BUT according toe the online solver it should be X=\left(\begin{matrix} 1\\3\end{matrix}\right)
    I have tried everything so I need someone's help, WHAT AM I DOING WRONG?
    Thank you!
    Please post the original question exactly as it's worded.
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    Re: Homogenous system of linear equations.

    Consider a female bird population with two age groups: adults and juveniles. The birds remain juvenile for one year and then become adults. The adult survival rate is 2/5 (that is, 2/5 of the adult birds survive from one year to the next), the juvenile survival rate is 1/5 (that is, the proportion of juveniles still alive and grown up the next year), and the reproduction rate is 3 (that is, each adult bird will have 3 offspring the following year).
    a0 = 56, and j0 = 40. Find an exact expression for a{k} and {jk} (as functions
    of k).
    So I have my A as I stated before, and I know how to do it, I am just having huge problems trying to find the P matrix to complete the problem.
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    Re: Homogenous system of linear equations.

    are you trying to find a P that diagonalizes A?

    in any case, your original matrix is in error. the equation I-A = 0 is:

    \begin{bmatrix}\frac{3}{5}&\frac{-1}{5}\\-3&1 \end{bmatrix}\begin{bmatrix}x\\y\end{bmatrix} = \begin{bmatrix}0\\0\end{bmatrix}

    which leads to x - \frac{1}{3}y = 0 , so an eigenvector is (1,3).
    Last edited by Deveno; October 1st 2011 at 06:21 PM.
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