My understanding of this part of linear algebra is somewhat clouded. I fail to see how these two problems are related and/or how to prove them:
1) Show that if $A$ is a linear transformation with $A^2 - A + 1 = 0$, then $A$ is invertible.
2) If $A$ and $B$ are linear transformations on the same vector space and $AB = 1$. Prove that if A has exactly one $right\space inverse\space B$, then A is invertible. (Consider $BA + B - 1$)
My Attempt:
1) Show that if $Ax = 0$, then $x$ must be $0$. i.e. $A^2 - A + 1 = 0$ implies $A^2 + 1 = A$, implies $(A^2 + 1)x = Ax$ imples $A(A(x)) + x = A(x)$ imples $A(0) + x = 0$. Since A is linear ($A(0) = 0$), $x = 0$.
2) No idea.
Hint
$$A^2-A+1=0\iff A(1-A)=1$$ and
$$A(BA+B-1)=A+1-A=1=AB$$