IAB is invertible if and only if IBA is invertible
In this entry $A$ and $B$ are endomorphisms^{} of a vector space^{} $V$. If $V$ is finite dimensional, we may choose a basis and regard $A$ and $B$ as square matrices^{} of equal dimension^{}.
Theorem^{}  Let $A$ and $B$ be endomorphisms of a vector space $V$. We have that

1.
$IAB$ is invertible^{} (http://planetmath.org/LinearIsomorphism) if and only if $IBA$ is invertible, and moreover

2.
$IAB$ is injective^{} if and only if $IBA$ is injective.
Proof :

1.
Suppose that $IAB$ is invertible. We shall prove that $B{(IAB)}^{1}A+I$ is the inverse^{} of $IBA$. In fact
$\left(IBA\right)\left(B{(IAB)}^{1}A+I\right)$ $=$ $B{(IAB)}^{1}A+IBAB{(IAB)}^{1}ABA$ $=$ $B\left({(IAB)}^{1}AB{(IAB)}^{1}\right)A+IBA$ $=$ $B\left((IAB){(IAB)}^{1}\right)A+IBA$ $=$ $BA+IBA$ $=$ $I$ A similar^{} computation shows that $\left(B{(IAB)}^{1}A+I\right)\left(IBA\right)=I$, i.e. $IBA$ is invertible.
Exchanging the roles of $A$ and $B$ we can prove the ”if” part. So $IAB$ is invertible if and only if $IBA$ is invertible.

2.
Let us first recall that a linear map between vector spaces is invertible if and only if its kernel $\mathrm{ker}$ is the zero vector (see this page (http://planetmath.org/KernelOfALinearTransformation)).
Suppose $IAB$ is not injective, i.e. there exists $u\ne 0$ such that $(IAB)u=0$. Then
$$(IBA)Bu=B(IAB)u=0$$ i.e. $Bu\in \mathrm{ker}(IBA)$. Notice that $Bu\ne 0$ because $u=ABu$ (by definition of $u$), so $IBA$ is also not injective.
Similarly, if $IBA$ is not injective then $IAB$ is not injective. $\mathrm{\square}$
Remark  It is known that for finite dimensional vector spaces a linear endomorphism is invertible if and only if it is injective. This does not remain true for infinite dimensional spaces, hence 1 and 2 are two different statements.
0.1 Comments
The result stated in 1 can be proven in a more general context — If $A$ and $B$ are elements of a ring with unity, then $IAB$ is invertible if and only if $IBA$ is invertible. See the entry on techniques in mathematical proofs, in which this result is proven using several different techniques.
This entry is based on http://planetmath.org/?op=getmsg&id=5088this discussion on PM.
Title  IAB is invertible if and only if IBA is invertible 

Canonical name  IABIsInvertibleIfAndOnlyIfIBAIsInvertible 
Date of creation  20130322 14:44:43 
Last modified on  20130322 14:44:43 
Owner  asteroid (17536) 
Last modified by  asteroid (17536) 
Numerical id  16 
Author  asteroid (17536) 
Entry type  Theorem 
Classification  msc 16B99 
Classification  msc 15A04 
Classification  msc 47A10 
Related topic  TechniquesInMathematicalProofs 