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### $I-AB$ be invertible $\Leftrightarrow$ $I-BA$ is invertible [duplicate]

assume $A,B\in M_n(F)$ if $I-AB$ be invertible then how to prove $I-BA$ is invertible and how find inverse of $I-BA$ Thanks in advance
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### Proof If $AB-I$ Invertible then $BA-I$ invertible. [duplicate]

I have these problems : Proof If $AB-I$ invertible then $BA-I$ invertible. Proof If $I-AB$ invertible then $I-BA$ invertible. I think I solve it correctly, But I'm not so sure, I'll be glad to ...
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### Proof if $I+AB$ invertible then $I+BA$ invertible and $(I+BA)^{-1}=I-B(I+AB)^{-1}A$ [duplicate]

I have the following question : Proof if $I+AB$ invertible then $I+BA$ invertible and $(I+BA)^{-1}=I-B(I+AB)^{-1}A$ I managed to proof that $I+BA$ invertible My proof : We know that $AB$ and $BA$ ...
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### $I_m -AB$ ivertible if and only if $I_n-BA$ invertible
Let $A$ and $B$ be $m\times n$ and $n\times m$ matrices respectively. Prove that if $\lambda$ is a non-zero eigenvalue of $AB$ then it is also an eigenvalue of $BA$ Prove that $I_m-AB$ is ...