This question already has an answer here:
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 ??
but how the expression on right hand side equal to $1$.
any hint ??