If a matrix A is such that
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

$3 A^{3}+2 A^{2}+5 A+I=0$

$\Rightarrow 3 A^{3}+2 A^{2}+5 A=-I$

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

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

$\Rightarrow A^{-1}=-3 A^{2}-2 A-5 I$