Definitions of Addition, Subtraction, Multiplication, and Division between Matrices of Different Orders and its Application to the Definition of Composite Groups, Along with Illustrative Examples

Standard matrix operations in classical linear algebra strictly require dimensional compatibility, such as identical orders for addition or matching inner dimensions for multiplication. Currently, interactions between matrices of mismatched dimensions rely heavily on
ad-hoc algorithmic workarounds such as zero-padding, pooling, and stride mechanics or computationally expensive tensor expansions like Kronecker products. These rigid constraints
create a theoretical gap when modeling modern computational operations, where small kernel matrices systematically interact with larger target matrices, often leaving indivisible
residual boundaries.
To bridge this gap, this paper introduces a highly generalized, natural, and profoundly
meaningful mathematical framework defining the block-wise operations of Addition (⊞),
Subtraction (⊟), Multiplication (⊠), and Division (⧸□) between square matrices of different
orders. By formally treating un-divisible rows and columns as preserved residual regions,
we establish a rigorous algebraic foundation for these cross-dimensional interactions without
resorting to artificial data manipulation. Furthermore, we investigate the algebraic properties of these operations, leading to the discovery of the Composite Group—a novel algebraic
structure where matrices of varying dimensions can form a perfectly closed group, seamlessly
governed by a native dimensional absorption rule among multiple identity elements.