In this example, $A$ and $B$ are square matrices, and $A+B$ is nonsingular, i.e. $(A+B)x = 0$ has only the trivial solution $x = 0$.
Does this logically imply that $(A+cB)x = 0$ has only the trivial solution also (where $c$ is a non-zero scalar)?
I have proved that if $B$ is nonsingular, then $cB$ must also be nonsingular, as the $c$ scalar can be interpreted as a series of type II EROs which don't affect the linear independence of matrix rows, but I can't see a way to prove the above.
Counterexample: Let $A=I$, $B=I$, $c = -1$