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?

It's important to note that this set of matrices forms a group but it does NOT form a subgroup of the matrix group $GL_2(\mathbb{R})$ (the group we are most familiar with as being a matrix group - the group of invertible $2\times 2$ matrices) as no elements in this set have non-zero determinant. In particular, we are looking at a subset of $Mat(\mathbb{R},2)$ which is disjoint from $GL_2(\mathbb{R})$.
The identity of the group will then be the matrix $\pmatrix{\frac{1}{2}&\frac{1}{2}\\ \frac{1}{2}&\frac{1}{2}}$ and the inverse of the element $\pmatrix{a&a\\ a&a}$ will be $\dfrac{1}{4}\pmatrix{a^{-1}&a^{-1}\\ a^{-1}&a^{-1}}$ (you should check this).
