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UPSC NDA Math Model Paper 10

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Question : 71 of 120

Marks: +1, -0

If a matrix

A

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

A^{-1}

equal to?

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

Solution:

Given, a matrix

A

is such that

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

Consider,

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

Pre - Multiply the above polynomial by

A^{-1}

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

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

\Rightarrow 3A^{-1}AA^{2}+2A^{-1}AA+5A^{-1}A+A^{-1}I=0

We know that

A^{-1}A=I

and

A^{-1}I=A^{-1}

\Rightarrow 3IA^{2}+2IA+5I+A^{-1}=0

\Rightarrow 3A^{2}+2A+5+A^{-1}=0

\Rightarrow 3A^{2}+2A+5+A^{-1}=0

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

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