If a matrix $A$ is invertible then can we prove that $(I+DA^{-1})$ is also invertible. Or, what are the necessary and sufficient condition for it to be invertible? Given that $D$ is a small perturbation of $A$ and $I$ is the identity matrix.
2026-03-29 05:42:34.1774762954
Invertibility of matrix.
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If $D = A + B$, $I+D A^{-1} = 2 I + B A^{-1}$. So if $\| B A^{-1}\| < 2$ (where $\| \cdot \|$ is any submultiplicative matrix norm), $I + D A^{-1}$ is invertible.