Question

If A satisfies the equation $x^3-5x^2+4x+\lambda =0$ then A⁻¹ exists if
(a) $\lambda \ne 1$
(b) $\lambda \ne 2$
(c) $\lambda \ne -1$
(d) $\lambda \ne 0$

Answer 1

$$\begin{gathered} \left(\mathrm{d}\right)\mathrm{\lambda }\ne 0 \\ A\mathrm{satisfies}{x}^3-5x^2+4x+\lambda \mathit{}=0. \\ \Rightarrow A^3-5A^{\mathit{2}}+4A=-\lambda \\ \mathrm{Assuming}{A}^{-1}\mathrm{exists},\mathrm{we}\mathrm{get} \\ {A}^{-1}\left(A^3-5A^2+4A\right)=-\lambda A^{-1} \\ \Rightarrow A^2-5A+4=-A^{-1}\mathrm{\lambda } \\ \Rightarrow A^{-1}=\frac{-\left(A^2-5A+4\right)}{\lambda } \\ \mathrm{Thus},{\mathrm{A}}^{-1}\mathrm{exists}\mathrm{if}\mathrm{\lambda }\ne 0. \end{gathered}$$