I've got a homework question that I've honestly no idea how to tackle. It goes as:
Let $A$, $B$ be real $n × n$ matrices such that the complex matrix $C = A + iB$ is
invertible. By considering $\det(A+\lambda B)$ as a function of $\lambda$, show that the matrix $A+\lambda B$
is invertible for some real number $\lambda$.
How do I consider $\det(A+\lambda B)$ as a function of $\lambda$?
Any help/hints would be greatly appreciated. Thanks.
Best Answer
Hint:
Well, for one thing, $\det (A+\lambda B)$ is indeed a function of $\lambda$, a polynomial in $\lambda$ in fact. Because it is a polynomial, you can tell a lot about the roots of the function (if a polynomial satisfies $p(x)=0$ for all $x\in \mathbb R$, what can you say about $p$?)