My text says$ \left\{\begin{pmatrix}a&a\\a&a\end{pmatrix}:a\ne0,a\in\mathbb R\right\}$ forms a group under matrix multiplication.
Solution 1:
The identity element in this group is $\pmatrix{\tfrac12&\tfrac12\\ \tfrac12&\tfrac12}$, not $I_2$.
Solution 2:
Call that matrix $(a)$ and you get $(a)(b)=(2ab)$ thus it's closed under multiplication and $(1/2)$ is its unit.