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.
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.
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.