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Question
If \[A^2 - A + I = 0\], then the inverse of A is __________ .
Options
A−2
A + I
I − A
A − I
Solution
I − A
\[\text{ Given: }A^2 - A + I = O\]
\[ A^{- 1} \left( A^2 - A + I \right) = A^{- 1} O ............. \left[\text{ multiplying both sides by }A^{- 1} \right]\]
\[ \Rightarrow \left( A^{- 1} A^2 \right) - \left( A^{- 1} A \right) + A^{- 1} I = O ...............\left[ \because A^{- 1} O = O \right]\]
\[ \Rightarrow A - I + A^{- 1} = O ...............\left[ \because A^{- 1} I = A^{- 1} \right]\]
\[ \Rightarrow A^{- 1} = I - A\]
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