Question

If for the matrix A, A³ = I, then A⁻¹ =
(a) A²
(b) A³
(c) A
(d) none of these

Answer 1

$$\begin{gathered} \left(\mathrm{a}\right)A^2 \\ \mathrm{Given}:\hspace{0.17em}{A}^3=I \\ \Rightarrow A^3{A}^{-1}=I{A}^{-1}\mathit{}\mathit{}\mathit{}\left[\mathrm{Multiplying}\mathrm{both}\mathrm{sides}\mathrm{by}{A}^{\mathit{-}\mathit{1}}\right] \\ \mathit{\Rightarrow }{A}^{\mathit{2}}\mathit{=}{A}^{\mathit{-}\mathit{1}} \end{gathered}$$