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Math Help - Finding the eigenvalues of a 3x3 matrix

  1. #1
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    Finding the eigenvalues of a 3x3 matrix

    Is there a method for determining the eigenvalues of a 3x3 matrix? The only examples my book does are already in upper triangular form, so det(1-tI) isn't exactly difficult (just multiply the diagonal). Any help on this? I've provided an example below!

    \left(\begin{array}{ccc}0&-2&-3\\-1&1&-1\\2&2&5\end{array}\right)
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  2. #2
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    ...
    Last edited by Unenlightened; July 30th 2009 at 03:06 PM.
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  3. #3
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    I would think that you already know that to find the eigenvalues of any matrix, A, you find the solutions to the equation det(A- \lambdaI)= 0.

    Here, A- \lambda I= <br />
\left(\begin{array}{ccc}-\lambda &-2 & -3\\-1&1-\lambda&-1\\2&2&5-\lambda\end{array}\right)<br />
.

    So the equation is \left|A- \lambda I\right|= \left|\begin{array}{ccc}-\lambda &-2 & -3\\-1&1-\lambda&-1\\2&2&5-\lambda\end{array}\right|= 0.


    It is probably simplest to expand that on the first column:
    \left|\begin{array}{ccc}-\lambda &-2 & -3\\-1&1-\lambda&-1\\2&2&5-\lambda\end{array}\right| = -\lambda\left|\begin{array}{cc}1-\lambda & -1 \\ 2  & 5-\lambda\end{array}\right| + \left|\begin{array}{cc}-2 & -3 \\ 2 & 5-\lambda\end{array}\right| + 2\left|\begin{array}{cc}-2 & -3 \\ 1-\lambda & -1 \end{array}\right|= 0.
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  4. #4
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    That's what I was afraid it was. It just seems so messy to me... Oh well!
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