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Question
If A satisfies the equation \[x^3 - 5 x^2 + 4x + \lambda = 0\] then A-1 exists if _____________ .
Options
\[\lambda = 1\]
\[\lambda \neq 2\]
\[\lambda \neq -1\]
\[\lambda \neq 0\]
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Solution
\[\lambda \neq 0\]
A satisfies \[ x^3 - 5 x^2 + 4x + \lambda = 0 . \]
\[ \Rightarrow A^3 - 5 A^2 + 4A = - \lambda\]
\[\text{ Assuming }A^{- 1}\text{ exists, we get }\]
\[ A^{- 1} \left( A^3 - 5 A^2 + 4A \right) = - \lambda A^{- 1} \]
\[ \Rightarrow A^2 - 5A + 4 = - A^{- 1} \lambda \]
\[ \Rightarrow A^{- 1} = \frac{- \left( A^2 - 5A + 4 \right)}{\lambda}\]
\[\text{ Thus, }A^{- 1}\text{ exists if } \lambda \neq 0 .\]
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