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Math Help - Determinants & Matrix Inverses

  1. #1
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    Determinants & Matrix Inverses

    If A and B are n by n, AB = -BA, and n is odd, show that either A or B are not invertible.

    Show that no 3 x 3 matrix A exists such that A^2 + I = 0.

    Any help is appreciated.
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  2. #2
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    Quote Originally Posted by BrownianMan View Post
    If A and B are n by n, AB = -BA, and n is odd, show that either A or B are not invertible.

    Show that no 3 x 3 matrix A exists such that A^2 + I = 0.

    Any help is appreciated.

    Since \det(AB)=\det(A)\det(B) , we get AB=-BA\Longrightarrow \det(AB)=\det(-BA)=\det(-B)\det(A)=(-1)^n\det(B)\det(A).

    Now suppose both matrices are invertible (i.e., their determinant is non-zero), and get a huge contradiction.

    Tonio
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  3. #3
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    You may have another constraint on your matrices you haven't mentioned yet - the matrix (where i=square root of -1):

    i,0,0
    0,i,0
    0,0,i

    satisfies the 2nd equation. If you demand that the matrix eigenvalues are real, the argument Tonio mentions should work.
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