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

Answer 1

None of these

\[3 A^3 + 2 A^2 + 5A + I = 0\]

\[ \Rightarrow 3 A^3 + 2 A^2 + 5A = - I\]

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

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

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