Question

Let A be a square matrix such that $A^2-A+I=O$, then write $A^{-1}$ interms of A.

Answer 1

$$\begin{gathered} \mathrm{Given}:A^2-A+I=O \\ {A}^{-1}\left(A^2-A+I\right)=A^{-1}O(\mathrm{Pre}-\mathrm{multiplying}\mathrm{both}\mathrm{sides}\mathrm{because}{A}^{-1}\mathrm{exists}) \\ \left(A^{-1}{A}^2\right)-\left(A^{-1}A\right)+A^{-1}I=O(A^{-1}O=O) \\ \Rightarrow A-I+A^{-1}=O(A^{-1}I=A^{-1}) \\ \Rightarrow A^{-1}=I-A \end{gathered}$$