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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- Oct 1st 2008, 09:51 PMsquarerootof2linear 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. - Oct 1st 2008, 10:39 PMwisterville
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. - Oct 2nd 2008, 07:43 AMThePerfectHacker
Define the function $\displaystyle f(c) = \det (A+cB)$. Notice that this function is a polynomial of degree $\displaystyle n$. Therefore by the fundamental theorem of algebra there exists a complex solution to $\displaystyle f(c) = 0$. And so $\displaystyle A+cB$ is not invertible.