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If A is a square matrix such that _{$(A - 2 I) (A + I) = 0$} ,then _{$A^{- 1} =$}

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_{$\frac{A - I}{2}$}

_{$\frac{A + I}{2}$}

_{$2 (A - I)$}

_{$2 A + I$}

answer is A.

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Detailed Solution

$(A - 2 I) (A + I) = 0$

$A A - 2 A I + A I - 2 I = 0 (∵ A I = A)$

$A A - A I = 2 I$

$\Rightarrow A (\frac{A - I}{2}) = I ∵ \frac{A - I}{2} = A^{- 1}$

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