A) \[\lambda \ne 1\]
B) \[\lambda \ne 2\]
C) \[\lambda \ne -1\]
D) \[\lambda \ne 0\]
Correct Answer: D
Solution :
Since, \[A\] satisfies the equation \[{{x}^{3}}-5{{x}^{23}}+4x+\lambda =O\] \[\Rightarrow \] \[{{A}^{3}}-5{{A}^{2}}+4A+\lambda I=O\] \[\Rightarrow \] \[A(-{{A}^{2}}+5A-4I)=\lambda I\] \[\Rightarrow \] \[A\left\{ \frac{1}{\lambda }(-{{A}^{2}}+5A-4I \right\}=I,\]if\[\lambda \ne 0\] Hence,\[{{A}^{-1}}\]exists and is equal to \[\frac{1}{\lambda }-({{A}^{2}}+5A-4I),\]if\[\lambda \ne 0\].
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