If for a square matrix A, A^2-A+I=0, then A^ -1 equals - Vidyadip

MCQ

If for a square matrix $A, A^2-A+I=0$, then $A^{-1}$ equals

A
$A$
B
$A+1$
C
$1- A$
D
$A-1$

✓

Answer

We have, $A^2-A+I=O$

Pre-multiplying with $A^{-1}$ on both sides, we get
$
\begin{array}{l}
\left(A^{-1} A\right) \cdot A-A^{-1} \cdot A+A^{-1} \cdot I=A^{-1} \cdot O \\
\Rightarrow \quad I \cdot A-I+A^{-1}=O \\
\Rightarrow \quad A^{-1}=-(A-I)=I-A
\end{array}
$

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