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