I stumbled upon this question that I would like to ask about:
Let $A$ and $B$ be $n\times n$ matrices over $\mathbb R$ where $B$ is an invertible matrix. How do you show that there exists some $\lambda \in \mathbb R$ such that $A+\lambda B $ is invertible?
Do I have to split this into the two cases where (i) $A$ is invertible, and (ii) $A$ is not invertible and then make an argument about the sum above? I want to use some determinant rule, perhaps it would help, but since this is a sum, I can't simply apply it; And I think I remember that only the product of two invertible matrices is again an invertible matrix, but for sums it is not so clear.