A² – A + I = 0 then the inverse of A

The correct option is I - A.

We are given the matrix equation:
$$A^2\&\#x2212;A+I=0$$ where $I$ is the identity matrix and $0$ is the zero matrix.

To find the inverse of the matrix $A$ (denoted as $A^{\&\#x2212;1}$), we can rearrange the terms to isolate the identity matrix $I$ on one side of the equation:

$$I=A\&\#x2212;A^2$$

Next, we factor out the matrix $A$ from the right-hand side. Since matrix multiplication is distributive over subtraction, we can write:

$$I=A(I\&\#x2212;A)$$

By definition, if two square matrices $X$ and $Y$ satisfy the relation $XY=I$, then $Y$ is the multiplicative inverse of $X$ (i.e., $X^{\&\#x2212;1}=Y$).

Comparing $A(I\&\#x2212;A)=I$ to this definition, we identify $X=A$ and $Y=I\&\#x2212;A$. Therefore, the inverse of $A$ is:

$$A^{\&\#x2212;1}=I\&\#x2212;A$$