Assume matrices $A$, $I-A$, $I-A^{-1}$ are all non-singular. Show that $$(I-A)^{-1} + (I-A^{-1})^{-1} = I$$
2026-04-24 22:45:28.1777070728
Non singular matrices
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$$\color{red}{(I-A)}\color{blue}{(I-A^{-1})}=I-A-A^{-1}+I = \color{red}{(I-A)}+\color{blue}{(I-A^{-1})}$$ Hence multiplying with the inverses:
first with $\color{red}{(I-A)^{-1}}$ on the left $$\color{blue}{I-A^{-1}}=I+\color{red}{(I-A)^{-1}}\color{blue}{(I-A^{-1})}$$ then with $\color{blue}{(I-A^{-1})^{-1}}$ on the right $$I=\color{blue}{(I-A^{-1})^{-1}}+\color{red}{(I-A)^{-1}}$$