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On matrices

Let A, B be two 3 \times 3 matrices with real entries. Prove that

A - \left ( A^{-1} +\left ( B^{-1} - A \right )^{-1} \right )^{-1} = ABA

provided all the inverses appearing on the left hand side exist.

(Vojtech Jarnik / 2nd Category/ 2015)

Solution

Let A, B be elements of an arbitrary associative algebra with unit. Then:

\begin{aligned} \left ( A^{-1} +\left ( B^{-1} - A \right )^{-1} \right )^{-1} &= \left ( A^{-1} \left ( B^{-1} - A \right )\left ( B^{-1} - A \right )^{-1} + A^{-1} A \left ( B^{-1} - A \right )^{-1} \right )^{-1} \\ &=\left ( A^{-1} \left ( \left ( B^{-1} - A \right ) +A \right )\left ( B^{-1} -A \right )^{-1} \right )^{-1} \\ &= \left ( A^{-1} B^{-1} \left ( B^{-1} -A \right )^{-1} \right )^{-1}\\ &= \left ( B^{-1}-A \right ) BA \\ &= A - ABA \end{aligned}

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