I don't know what you mean, but for Matrix-Products you should use Dot : a.b
a = {{0, 1}, {1, 1}}
MatrixForm[a]
a.a (* your a^2 or a*a * )
v = {{2, 2}, {1 + Sqrt[5], 1 - Sqrt[5]}}
Inverse[v].a.v
(Inverse[v].a.v). (Inverse[v].a.v)
v. (Inverse[v].a.v).(Inverse[v].a.v). Inverse[v]//FullSimplify
I hope this helps