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  4. Let A be a non-singular symmetric matrix of order 3 . If AT=A2-I , then (A - I)- 1 is equal to

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Q. Let $A$ be a non-singular symmetric matrix of order $3$ . If $A^{T}=A^{2}-I$ , then $\left(A - I\right)^{- 1}$ is equal to

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Solution:

$\because A^{T}=A$ (as $A$ is symmetric)
$\therefore A=A^{2}-I\Rightarrow A^{2}-A=I$
$\Rightarrow I=A\left(A - I\right)=\left(A - I\right)A$
$\Rightarrow \left(A - I\right)^{- 1}=A$