Necessary and/or sufficient conditions for $A+B$ to be invertible Let $A$ and $B$ be two $n\times n$ real invertible matrices. Are there necessary and/or sufficient conditions (involving only $A$ and $B$ separately, not $(A+B)$ iteself) for $A+B$ to be invertible? We exclude, for example, the trivial conditions $\text{det}(A+B)\neq 0$ and $\text{det}((A+B)^T)\neq 0$.
Sufficient condition: If $A$ and $B$ are positive definite, $A+B$ is positive definite and therefore invertible. Also if $A$ is positive semi-definite and $B$ is positive definite, $A+B$ is positive definite and invertible.
No, because two invertible matrices A and B, when added, may end up non-invertible.