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Math Help - linear algebra question

  1. #1
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    linear algebra question

    Let A and B be nxn matrices over the field of complex numbers.
    How would i show that if B is invertible, then there exists a scalar c in C such that A+cB is not invertible?

    i was examining det(A+cB) as given by hints but could not reach a solution.
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  2. #2
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    Hello,

    Use det(XY)=det(X)det(Y).
    Find c such that det(AB^{-1}+cE)=0. -c is the eigenvalue of AB^{-1}.

    Bye.
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  3. #3
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    Quote Originally Posted by squarerootof2 View Post
    i was examining det(A+cB) as given by hints but could not reach a solution.
    Define the function f(c) = \det (A+cB). Notice that this function is a polynomial of degree n. Therefore by the fundamental theorem of algebra there exists a complex solution to f(c) = 0. And so A+cB is not invertible.
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