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.