Answer
No
Work Step by Step
No, it is not because the sum of the inverses of the two matrices is not invertible
the reason in the following
Since $A$ and $B$ are invertible matrices, then
$A+B=AB^{-1}B+AA^{-1}B=A(B^{-1}+A^{-1})B$
and then $(A+B)^{-1}=(A(B^{-1}+A^{-1})B)^{-1}=((B^{-1}+A^{-1})B)^{-1}A^{-1}=B^{-1} (B^{-1}+A^{-1})^{-1} A^{-1}$
for example, if $A=I$ and $B=-I$ , then $A^{-1}=I$ and $B^{-1}=-I$ , thus $B^{-1}+A^{-1}=O$ , thus the matrix $B^{-1}+A^{-1}$ is singular and then the matrix $B^{-1}+A^{-1}$ is not invertible
