I stumbled upon this question that I would like to ask you about:
Let $A$ be a $n\times n$ matrix $(\mathbb R)$ and $B$ an invertible Matrix of size $n$ with real coefficients.
I need to show that there exists some $\lambda \in \mathbb R$ such that $$A+\lambda B $$ is invertible.
So do I have to split this into two cases?
i) $A$ is invertible ii) $A$ is not invertible
and make an argument for 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 to remember that only the product of two invertible matrices is again an invertible matrix, but for sums it is not so clear.
Any help would be greatly appreciated!