Question

If A satisfies the equation \[x^3 - 5 x^2 + 4x + \lambda = 0\] then A^-1 exists if _____________ .

Options

• \[\lambda = 1\]

• \[\lambda \neq 2\]

• \[\lambda \neq -1\]

• \[\lambda \neq 0\]

Answer 1

\[\lambda \neq 0\]

A satisfies \[ x^3 - 5 x^2 + 4x + \lambda = 0 . \]

\[ \Rightarrow A^3 - 5 A^2 + 4A = - \lambda\]

\[\text{ Assuming }A^{- 1}\text{ exists, we get }\]

\[ A^{- 1} \left( A^3 - 5 A^2 + 4A \right) = - \lambda A^{- 1} \]

\[ \Rightarrow A^2 - 5A + 4 = - A^{- 1} \lambda \]

\[ \Rightarrow A^{- 1} = \frac{- \left( A^2 - 5A + 4 \right)}{\lambda}\]

\[\text{ Thus, }A^{- 1}\text{ exists if } \lambda \neq 0 .\]