What is the name of the algebra that is under the multiplication of matrices of different sizes? I understand that Matrix Group(Linear Group) is a group for invertible square matrices under multiplication, Matrix Ring is a ring for square matrices under both multiplication and addition. But they are all limited to special kinds of matrices. What I'm looking for is the algebra for multiplication of matrices of different sizes. Can I have the name of it please?
 A: The algebraic structure you are looking for is called a category.
The easiest way to see this is that matrices (with coefficients in for example $\mathbb{R}$) are in bijective correspondence with linear maps between finite dimensional vector spaces over $\mathbb{R}$. So this algebraic structure you are thinking about is simply a category of finite dimensional vector spaces over $\mathbb{R}$. Matrices are the morphisms of this category, and the matrix multiplication is the composition operation.

Let's be more precise. To define a category (let's call it $\mathcal{C}$), we have to provide a set of objects, for wich we will take 
$$\operatorname{obj}(\mathcal{C})=\{\mathbb{R}^n \mid n \in \mathbb{N} \}$$
We also have to provide a set of morphisms for each pair of objects. So for the pair $(\mathbb{R}^n, \mathbb{R}^m)$ we take the set of all linear maps between thes 2 spaces, which is equivalent to the set of all $m$ by $n$ matrices.
$$\operatorname{Hom}_{\mathcal{C}}(\mathbb{R}^n, \mathbb{R}^m) = \mathbb{R}^{m\times n}$$
In a category you then need to define a composition operator, which is in our case the matrix multiplication or equivalently the composition of linear maps.
You can check for yourself that this indeed satisfies the necessary properties to be a category (associativity and identity).

We can go further: there is also an algebraic structure that describes both the addition and multiplication of general matrices. This structure is called an abelian category.
