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Math Help - How to prove: A,B are orthogonal matrices, if det(A)+det(B)=0, then A+B is singular.

  1. #1
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    How to prove: A,B are orthogonal matrices, if det(A)+det(B)=0, then A+B is singular.

    Thanks a lot!!!
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    Forum Admin topsquark's Avatar
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    Quote Originally Posted by seekor View Post
    Thanks a lot!!!
    Where is the question? seekor, please don't delete posts once you have the answer. Someone else might be able to benefit from them as well.

    -Dan
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    Quote Originally Posted by topsquark View Post
    The other thread has been locked. The question is:


    If det(A) + det(B) = 0 then
    det(A + B) = det(A) + det(B) = 0

    Thus the matrix A + B is singular because it has a zero determinant.

    -Dan
    In general, det(A + B) \ne det(A) + det(B).

    Example: A = \begin{pmatrix} 1 & 0 \\ 0 & 0  \end{pmatrix},\ B = \begin{pmatrix} 0 & 0 \\ 0 & 1  \end{pmatrix},\ det(A) = det(B) = 0,\  det(A+B) = 1.

    For square matrices, det(AB) = det(A)det(B).
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  4. #4
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by JakeD View Post
    In general, det(A + B) \ne det(A) + det(B).

    Example: A = \begin{pmatrix} 1 & 0 \\ 0 & 0  \end{pmatrix},\ B = \begin{pmatrix} 0 & 0 \\ 0 & 1  \end{pmatrix},\ det(A) = det(B) = 0,\  det(A+B) = 1.

    For square matrices, det(AB) = det(A)det(B).
    (sigh) I thought there was something fishy going on there. I guess I was tired last night! Thanks for the spot.

    -Dan
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