If a matrix A is such that $3 A^{3} + 2 A^{2} + 5 A + I = 0$ , then $A^{- 1}$ is equal to

- (3 A^{2} + 2 A + 5)

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3 A^{2} + 2 A + 5

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3 A^{2} - 2 A - 5

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none of these

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Solution

The correct option is B $- (3 A^{2} + 2 A + 5)$
Given, $3 A^{3} + 2 A^{2} + 5 A + I = O$
$\Rightarrow 3 A^{3} A^{- 1} + 2 A^{2} A^{- 1} + 5 A A^{- 1} + I A^{- 1} = O$
$\Rightarrow 3 A^{2} + 2 A + 5 I + A^{- 1} = O$
$\Rightarrow A^{- 1} = - (3 A^{2} + 2 A + 5 I)$
Hence, option 'A' is correct.

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