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If $A=\left[\begin{array}{cc}2 & -1 \\ -1 & 2\end{array}\right]$ such that $A^2-4 A+3 I=0$, where $\mathrm{I}$ is a unit matrix of order 2 , then $A^{-1}$ is
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The correct answer is:
$\frac{1}{3}\left[\begin{array}{ll}2 & 1 \\ 1 & 2\end{array}\right]$
$\begin{aligned} & A^2-4 A+3 I=0 \Rightarrow A(4 I-A)=3 I \Rightarrow A\left(\frac{4 I-A}{3}\right)=I \\ & \Rightarrow A^{-1}=\frac{4 I-A}{3}=\frac{1}{3}\left[\begin{array}{ll}2 & 1 \\ 1 & 2\end{array}\right]\left[\text { as } A A^{-1}=I\right]\end{aligned}$
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