If a matrix \( A \) is non-singular matrix of order \( 2\times 2 \) such that \( 3A^{3}+2A^{2}+5A+I=O \), where \( I \) is \( 2\times 2 \) identity matrix and \( O \) is \( 2\times 2 \) zero matrix, then \( A^{-1} \) is
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \( -\left[3A^{2}+2A+5I\right] \)
Since A is non-singular, \( A^{-1} \) exists. Multiplying the given relation by \( A^{-1} \),
\( \displaystyle A^{-1}\left(3A^{3}+2A^{2}+5A+I\right)=O \)
\( \displaystyle 3A^{2}+2A+5I+A^{-1}=O \)
\( \displaystyle A^{-1}=-\left(3A^{2}+2A+5I\right) \)