Question

If A⁵ = O such that $A^n\ne I\mathrm{for}1\le n\le 4,\mathrm{then}{\left(I-A\right)}^{-1}$ equals
(a) A⁴
(b) A³
(c) I + A
(d) none of these

Answer 1

$$\begin{gathered} \left(\mathrm{d}\right)\mathrm{none}\mathrm{of}\mathrm{the}\mathrm{these} \\ I-A^5=\left(I-A\right)\left(I+A+A^2+A^3+A^4\right) \\ \text{Now,} \\ {A}^5=0 \\ \Rightarrow I=\left(I-A\right)\left(I+A+A^2+A^3+A^4\right) \\ \Rightarrow \frac{I}{\left(I-A\right)}=\left(I+A+A^2+A^3+A^4\right) \\ \Rightarrow {\left(I-A\right)}^{-1}=I+A+A^2+A^3+A^4 \end{gathered}$$