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If a matrix $A$ is such that
$3 A^{3}+2 A^{2}+5 A+I=0$
Then what is $A^{-1}$ equal to?
Options:
$3 A^{3}+2 A^{2}+5 A+I=0$
Then what is $A^{-1}$ equal to?
Solution:
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Verified Answer
The correct answer is:
$-\left(3 A^{2}+2 A+5\right)$
Let A bea matrix such that $3 \mathrm{~A}^{3}+2 \mathrm{~A}^{2}+5 \mathrm{~A}+\mathrm{I}=0$
Post multiply by $\mathrm{A}^{-1}$ on both the sides, we get $3 \mathrm{~A}^{3} \mathrm{~A}^{-1}+2 \mathrm{~A}^{2} \mathrm{~A}^{-1}+5 \mathrm{AA}^{-1}+\mathrm{IA}^{-1}=0$
$\Rightarrow 3 \mathrm{~A}^{2}+2 \mathrm{~A}+5 \mathrm{I}+\mathrm{A}^{-1}=0$
$\Rightarrow \mathrm{A}^{-1}=-\left(3 \mathrm{~A}^{2}+2 \mathrm{~A}+5 \mathrm{I}\right)$
Post multiply by $\mathrm{A}^{-1}$ on both the sides, we get $3 \mathrm{~A}^{3} \mathrm{~A}^{-1}+2 \mathrm{~A}^{2} \mathrm{~A}^{-1}+5 \mathrm{AA}^{-1}+\mathrm{IA}^{-1}=0$
$\Rightarrow 3 \mathrm{~A}^{2}+2 \mathrm{~A}+5 \mathrm{I}+\mathrm{A}^{-1}=0$
$\Rightarrow \mathrm{A}^{-1}=-\left(3 \mathrm{~A}^{2}+2 \mathrm{~A}+5 \mathrm{I}\right)$
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