Question

If $A^2-A+I=0$, then the inverse of A is
(a) A⁻²
(b) A + I
(c) I − A
(d) A − I

Answer 1

(c) I − A

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