Let A be a square matrix


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


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)

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

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

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

Leave a comment


Click here to get exam-ready with eSaral

For making your preparation journey smoother of JEE, NEET and Class 8 to 10, grab our app now.

Download Now