Solve this

Question:

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

(a) $A^{-2}$

(b) $A+1$

(c) $I-A$

(d) $A-1$

Solution:

(c) $I-A$

Given : $A^{2}-A+I=O$

$A^{-1}\left(A^{2}-A+I\right)=A^{-1} O \quad$ [multiplying both sides by $A^{-1}$ ]

$\Rightarrow\left(A^{-1} A^{2}\right)-\left(A^{-1} A\right)+A^{-1} I=O \quad\left[\because A^{-1} O=O\right]$

$\Rightarrow A-I+A^{-1}=O \quad\left[\because A^{-1} I=A^{-1}\right]$

$\Rightarrow A^{-1}=I-A$

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