Answer
No
Work Step by Step
No, because the sum of the inverses of the matrices $A$ and $B$ must be invertible
and in the following, we indicate the reason
Since $A$ and $B$ are invertible matrices, then
$A+B=AB^{-1}B+AA^{-1}B=A(B^{-1}+A^{-1})B$
and then $(A+B)^{-1}=(A(B^{-1}+A^{-1})B)^{-1}=((B^{-1}+A^{-1})B)^{-1}A^{-1}=B^{-1} (B^{-1}+A^{-1})^{-1} A^{-1}$
Example :- if $A=I$ and $B=-I $ , then $A^{-1}=I $and $B^{-1}=-I$ , Then $B^{-1}+A^{-1}=O$ and the matrix $B^{-1}+A^{-1}$ is singular , thus in this example the matrix $B^{-1}+A^{-1}$ is not invertible
