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Math Help - Finding all 2 by 2 matrices that have eigenvalues x and y

  1. #1
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    Post Finding all 2 by 2 matrices that have eigenvalues x and y

    I have been pulling my hair in solving the following problem and I have reached to the deadline. If anyone can give me an idea of how can I solve the problem, please help me...

    Problem: Find all the 2 by 2 matrices that have x and y as eigenvalues.

    You may replace the values of x and y, say by 4 and 5 to show me an example of how the problem can be solved...

    Thank you very much for your time... It means a lot to me... Hope to get some idea of how the problem can be solved....
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  2. #2
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    Quote Originally Posted by precious_pearl13 View Post
    I have been pulling my hair in solving the following problem and I have reached to the deadline. If anyone can give me an idea of how can I solve the problem, please help me...

    Problem: Find all the 2 by 2 matrices that have x and y as eigenvalues.

    You may replace the values of x and y, say by 4 and 5 to show me an example of how the problem can be solved...

    Thank you very much for your time... It means a lot to me... Hope to get some idea of how the problem can be solved....
    Just reverse engineer the problem.

    If x and y are eigenvalues, then the characteristic polynomial is:

     (\lambda - x)(\lambda - y) = 0

    Since there are two eigenvalues you are expecting a matrix or order 2. Meaning your characteristic matrix is:

    \left( \begin{array}{ccc}<br />
a & b  \\<br />
c & d  \end{array} \right)

    You know the determinant of such a matrix must give yhou the characteristic polynomial, so:

     ad-bd = (\lambda - x)(\lambda - y)

    See where I'm going with this?
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