How can I prove that $(PQ + I_N)^{-1}P = P(QP + I_M)^{-1}$ knowing that we have two matrix $P_{N \times M}$ and $Q_{M \times N}$.
Thank you very much for help.
2026-04-07 23:35:44.1775604944
Matrix multiplication identity proof
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Assuming that both $PQ + I_N$ and $QP + I_M$ are invertible just multiply from the left with $(PQ + I_N)$ and from the right with $(QP + I_M)$ to obtain the equivalent statement $$ P(QP+I_M) = (PQ + I_N)P, $$ which is true.