Is the inverse operation on matrices distributive with respect to addition?
For example, is the following true: $$(A + B)^{-1} = A^{-1} + B^{-1}$$ If $\det(A) \ne 0$, $\det(B) \ne 0$, and $\det(A + B) \ne 0$.
Solution 1:
Is it true for $1\times 1$ matrices?
Solution 2:
Suppose that $(A+B)^{-1} = A^{-1}+B^{-1}$. Then,
$$\begin{align*} I &= (A+B)^{-1}(A+B) \\ &= (A^{-1}+B^{-1})(A+B) \\ &= A^{-1}A + A^{-1}B + B^{-1}A+B^{-1}B \\ &= 2I + A^{-1}B + B^{-1}A, \end{align*}$$ so $A^{-1}B+B^{-1}A = -I$ for all invertible $A$, $B$.
It should be easy to use this to construct a counterexample.