Let A be a square matrix

Question:

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

Solution:

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

$A^{-1}\left(A^{2}-A+I\right)=A^{-1} O$        (Pre - multiplying both sides because $A^{-1}$ exists)

$\left(A^{-1} A^{2}\right)-\left(A^{-1} A\right)+A^{-1} I=O$     $\left(A^{-1} O=O\right)$

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

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

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