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. 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.
But I can see $I\notin$ the set and so not a group.
Am I right?

If the structure be a group so it has an identity element, so for any matrix called $A$ we have: $$A:=\begin{pmatrix}
  a & a \\
 a & a \\
  \end{pmatrix}$$ for some $a\neq 1$ and if $$\text{id}_G=\begin{pmatrix}
  b & b \\
 b & b \\
  \end{pmatrix}, ~~b\neq 0$$ then $A\times\text{id}_G=\text{id}_G\times A=A$. Now $$A\times\text{id}_G=\begin{pmatrix}
  2ab & 2ab \\
 2ab & 2ab \\
  \end{pmatrix}$$ which should be equal to $A$ itself. So $a=2ab$ and since $a\neq 0$ so $b=0.5$.
