$A , B$ square matrices of size $n$ with real entries with $B$ invertible , the does $\exists c \in \mathbb R$ such that $\det (A+cB)=0$?

Question

Let $A$ be a $n \times n$ matrix with real entries and $B$ is an invertible $n \times n$ matrix with real entries ; then does there exist $c \in \mathbb R$ such that $\det(A+cB)=0$ ?

Answer 1

If $A$ is not invertible, just take $c=0$. If $A$ is invertible, then

$$\det (A + cB) = \det A \det (I + c A^{-1} B)$$

so it suffices to know given an invertible $M$, whether there is $c$ so that $I + cM$ is not invertible. Thus you are asking whether there is $x\neq 0$ such that

$$(I + cM)x = 0 \Leftrightarrow Mx = -\frac{1}{c} x. $$

(Note $c\neq 0$). Thus going back to your question, such a $c$ can be found if and only if $M=A^{-1} B$ has a real eigenvalue.