Question

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

Answer 1

\[\text{ Given: } A^2 - A + I = O\]
\[ A^{- 1} \left( A^2 - A + I \right) = A^{- 1} O .................(\text{Pre - multiplying both sides because }A^{- 1}\text{ exists})\]
\[\left( A^{- 1} A^2 \right) - \left( A^{- 1} A \right) + A^{- 1} I = O...................  ( A^{- 1} O = O)\]
\[ \Rightarrow A - I + A^{- 1} = O................... ( A^{- 1} I = A^{- 1} )\]
\[ \Rightarrow A^{- 1} = I - A\]