Sum A+cB of invertible matrices noninvertible?

  • Thread starter Grothard
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  • #1
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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?
 

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  • #2
D H
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Prove this theorem by construction. Hint: A non-invertible matrix has at least one zero eigenvalue.
 
  • #3
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So c = (-λ_a_k)/(-λ_b_k), thanks!
 

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