Sum A+cB of invertible matrices noninvertible?

1. Nov 11, 2011

Grothard

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. Nov 11, 2011

D H

Staff Emeritus
Prove this theorem by construction. Hint: A non-invertible matrix has at least one zero eigenvalue.

3. Nov 11, 2011

Grothard

So c = (-λ_a_k)/(-λ_b_k), thanks!