# Does any equation holds for inverse(A- B)?

is it possible to calculate inverse of A-B, i have calculated inverse of A and B already.

A and B are matrices

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possible duplicate of Inverse of the sum of matrices – Arturo Magidin Mar 30 '11 at 16:19

You can express $(A-B)^{-1} = -(B-A)^{-1}$ in terms of either matrix and the inverse of the other matrix as a formal power series; however, the convergence of these series are not guaranteed. $$\begin{eqnarray} (A-B)^{-1} &=& \left(A(I-A^{-1}B)\right)^{-1} \\ &=& \left(I-A^{-1}B\right)^{-1}A^{-1} \\ &=& +A^{-1} + A^{-1}BA^{-1} + A^{-1}BA^{-1}BA^{-1} + ... \\ &=& -B^{-1} - B^{-1}AB^{-1} - B^{-1}AB^{-1}AB^{-1} - ... \end{eqnarray}$$
The text passage contains a mathematical question about the inverse of the difference of two matrices, and provides a detailed answer in the form of a formal power series representation.
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# Does any equation holds for inverse(A- B)?

is it possible to calculate inverse of A-B, i have calculated inverse of A and B already.

A and B are matrices

-
possible duplicate of Inverse of the sum of matrices – Arturo Magidin Mar 30 '11 at 16:19

You can express $(A-B)^{-1} = -(B-A)^{-1}$ in terms of either matrix and the inverse of the other matrix as a formal power series; however, the convergence of these series are not guaranteed. $$\begin{eqnarray} (A-B)^{-1} &=& \left(A(I-A^{-1}B)\right)^{-1} \\ &=& \left(I-A^{-1}B\right)^{-1}A^{-1} \\ &=& +A^{-1} + A^{-1}BA^{-1} + A^{-1}BA^{-1}BA^{-1} + ... \\ &=& -B^{-1} - B^{-1}AB^{-1} - B^{-1}AB^{-1}AB^{-1} - ... \end{eqnarray}$$
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The text passage contains a mathematical question about the inverse of the difference of two matrices, and provides a detailed answer in the form of a formal power series representation.

TYPE: SINGLE_QUESTION_WITH_ANSWER