I wish to show that the subset $A$ of $GL(2 ,\Bbb R)$ consisting of matrices of the form
$$\begin{bmatrix} a &b\\0 &a \end{bmatrix}$$
where $a >0$ is a regular submanifold of dimension $2$. The easy way to show that it's a regular submanifold is to invoke Cartan's theorem. But I do not wish to use it. So I'll proceed as follows: first associate $2 \times2$ matrices in $A$ with elements of $\Bbb R^2$ in a natural way $$\begin{bmatrix} x &y\\0 &x \end{bmatrix} \longmapsto (x,y)$$
since $x >0$, $A$ is an open subset of $\Bbb R^2$ and therefore, $A$ is a smooth manifold of dimension $2$. But I'm not able to show $A$ is a regular submanifold. Any hints on how to proceed will be highly appreciated.
Let $U = \{(x,y) \in \Bbb R : x \ne 0 \}$. It suffices to note that the map $h:U \to A$ defined by $$ h(x,y) = \pmatrix{x & y\\0 & x} $$ is bijective and continuous with continuous inverse. That is, $h$ is a homeomorphism, so the map is indeed a topological embedding.
To see that this map is continuous, it suffices to note that the topology on $\Bbb R^{2 \times 2}$ is the product topology, and $A \subset GL(2,\Bbb R) \subset \Bbb R^{2 \times 2}$ has the associated subspace topology.