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Question
If a matrix A is such that \[3A^3 + 2 A^2 + 5 A + I = 0,\text{ then }A^{- 1}\] equal to _______________ .
Options
\[- \left( 3 A^2 + 2 A + 5 \right)\]
\[3 A^2 + 2 A + 5\]
\[3 A^2 - 2 A - 5\]
none of these
Solution
None of these
\[3 A^3 + 2 A^2 + 5A + I = 0\]
\[ \Rightarrow 3 A^3 + 2 A^2 + 5A = - I\]
\[ \Rightarrow A^{- 1} (3 A^3 + 2 A^2 + 5A) = - I A^{- 1} \]
\[ \Rightarrow 3 A^2 + 2A + 5I = - A^{- 1} \]
\[ \Rightarrow A^{- 1} = - 3 A^2 - 2A - 5I\]
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