Question

(c) Is it possible for an n × n matrix A to satisfy A3 = In without A being invertible? Explain

Answer 1

If $A^3=I$, then $A^3=AA^2=I$ which means $A$ and $A^2$ must be inverses of one another, so $(A^2)^{-1}=A$ and $A^{-1}=A^2$, so $A$ must be invertible.