Linked Questions

11
votes
3answers
11k views

$I-AB$ 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
6
votes
4answers
2k views

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 ...
5
votes
4answers
918 views

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 $BA$ ...
6
votes
1answer
2k views

Prove that if $I-BA$ is invertible then $I-AB$ is invertible [duplicate]

Prove that if $I-BA$ is invertible then $I-AB$ is invertible. Though I have found this question already posted and also it has some answers like Use:$(I−BA)(I+B(I−AB)^{−1}A)=I(I−BA)(I+B(I−AB)^{−1}...
3
votes
4answers
212 views

Invert of Matrix I-BA [duplicate]

Suppose $A$ and $B$ are two square Matrix. Let $I-AB$ be invertible. I would like to know why $I-BA$ is also invertible? Also what is invert of $I-BA$? Thanks.
2
votes
3answers
1k views

If $I-AB$ is invertible, then is $I-BA$ invertible? [duplicate]

If $A$, $B$ are square matrices and $I-AB$ is invertible how do I prove that $I-BA$ is invertible? This is exercise 8 of section 6.2 in Linear Algebra by Hoffman and Kunze. My thoughts. If $A$ and $...
3
votes
2answers
191 views

Inverse of $I+BA$ is $(I+BA)^{-1} = I-B(I+AB)^{-1}A$ [duplicate]

Assume $I+AB$ is invertible, prove then that $I+BA$ is invertible and $(I+BA)^{-1} = I-B(I+AB)^{-1}A$. My work: $(I+AB)(I+AB)^{-1} = I$ $B(I+AB)(I+AB)^{-1} = B $ $(I+BA)B(I+AB)^{-1} = B$ $(I+BA)...
0
votes
1answer
61 views

if $a,b$ are elements of a unital algebra $A,$ then $1-ab$ is invertible if and only if $1-ba$ is invertible. [duplicate]

if $a,b$ are elements of a unital algebra $A,$ then $1-ab$ is invertible if and only if $1-ba$ is invertible. because if $1-ab\ $ has inverse $x$ , then $1-ba\ $ has inverse $1+bxa$. but how ?? $$(...
0
votes
0answers
48 views

Characteristic values. [duplicate]

Let A and B be two $ n\times n$ matrcies over the field F.Prove that if (I-AB) is invertible, then (I-BA) is invertible and that $ (I-BA)^{-1}=I+B(I-AB)^{-1}A $ Using this result prove that AB and ...
3
votes
4answers
936 views

How to prove $I-BA$ is invertible [duplicate]

Show that $I-BA$ is invertible if $I-AB$ is invertible. And also, we have to prove that eigenvalues are same for $AB$ and $BA$ Till now, I used the equation $(I-AB)(I-AB)^{-1}=I$ which gives $(I-AB)...
3
votes
2answers
156 views

$I_m -AB$ 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 ...