If $A^2-A+I=0$, then the inverse of $A$ is:

Given the matrix equation:
$$A^2-A+I=0$$
To find the inverse $A^{-1}$, we multiply both sides of the equation by $A^{-1}$:
$$A^{-1}(A^2-A+I)=A^{-1}\cdot 0$$
Using the distributive property:
$$A^{-1}{A}^2-A^{-1}A+A^{-1}I=0$$
Since $A^{-1}A=I$ and $A^{-1}I=A^{-1}$:
$$A-I+A^{-1}=0$$
Rearranging the terms to solve for $A^{-1}$:
$$A^{-1}=I-A$$