Question

How to do ii) Deduce A^-1 = A^2 based on A^3=I

Answer 1

Multiply by $A^{-1}$ on the right of both sides of $A^3=I$, then use the fact that matrix multiplication is associative.

If $A^3=I$, then

$A^3A^{-1}=IA^{-1}$

$A^2(AA^{-1})=A^{-1}$

$A^2I=A^{-1}$

$\implies A^2=A^{-1}$