= Solution
Besides concatenation $*$, loops in $U(2)$ have pointwise multiplication
$$
(\alpha\mathbin{\cdot}\beta)(t)=\alpha(t)\beta(t).
$$
Continuity of matrix multiplication makes this a loop, and pointwise multiplication of homotopies shows that it is well-defined on homotopy classes. Both operations have the constant loop $e(t)=I$ as identity, and they satisfy the interchange law
$$
(\alpha*\beta)\mathbin{\cdot}(\gamma*\delta)
=(\alpha\mathbin{\cdot}\gamma)*(\beta\mathbin{\cdot}\delta).
$$
For homotopy classes $a,b$, the interchange law gives
$$
a\mathbin{\cdot}b
=(a*e)\mathbin{\cdot}(e*b)
=(a\mathbin{\cdot}e)*(e\mathbin{\cdot}b)
=a*b.
$$
It then gives
$$
a*b=a\mathbin{\cdot}b
=(e*a)\mathbin{\cdot}(b*e)
=(e\mathbin{\cdot}b)*(a\mathbin{\cdot}e)
=b*a.
$$
Thus loop concatenation is commutative on homotopy classes, and
$$
\boxed{\pi_1(U(2),I)\text{ is abelian}.}
$$
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