If a matrix A is such that 3 $A^{3} + 2 A^{2} + 5 A + I = 0, then A^{- 1}$ equal to
(a) $- (3 A^{2} + 2 A + 5)$
(b) $3 A^{2} + 2 A + 5$
(c) $3 A^{2} - 2 A - 5$
(d) none of these

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Solution

(d) none of these

$3 A^{3} + 2 A^{2} + 5 A + I = 0 \Rightarrow 3 A^{3} + 2 A^{2} + 5 A = - I \Rightarrow A^{- 1} (3 A^{3} + 2 A^{2} + 5 A) = - I A^{- 1} \Rightarrow 3 A^{2} + 2 A + 5 I = - A^{- 1} \Rightarrow A^{- 1} = - 3 A^{2} - 2 A - 5 I$

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