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Sum A+cB of invertible matrices noninvertible?

  1. Nov 11, 2011 #1
    If A and B are both invertible square matrices of the same size with complex entries, there exists a complex scalar c such that A+cB is noninvertible.

    I know this to be true, but I can't prove it. I tried working with determinants, but a specific selection of c can only get rid of one entry in A+cB, which is not enough since the matrices do not have to be triangular. How would one go about proving this theorem?
     
  2. jcsd
  3. Nov 11, 2011 #2

    D H

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    Prove this theorem by construction. Hint: A non-invertible matrix has at least one zero eigenvalue.
     
  4. Nov 11, 2011 #3
    So c = (-λ_a_k)/(-λ_b_k), thanks!
     
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