Question:
Let $A$ be a square matrix such that $A^{2}-A+I=O$, then write $A^{-1}$ interms of $A$.
Solution:
Given : $A^{2}-A+I=O$
$A^{-1}\left(A^{2}-A+I\right)=A^{-1} O$ (Pre - multiplying both sides because $A^{-1}$ exists)
$\left(A^{-1} A^{2}\right)-\left(A^{-1} A\right)+A^{-1} I=O$ $\left(A^{-1} O=O\right)$
$\Rightarrow A-I+A^{-1}=O \quad\left(A^{-1} I=A^{-1}\right)$
$\Rightarrow A^{-1}=I-A$