Question

If \[A^2 - A + I = 0\], then the inverse of A is __________ .

Options

• A⁻²

• A + I

• I − A

• A − I

Answer 1

I − A

\[\text{ Given: }A^2 - A + I = O\]

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

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

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

\[ \Rightarrow A^{- 1} = I - A\]