Matrix addition represents a fundamental binary operation within linear algebra defined strictly on conformable matrices sharing identical dimensions. The core principle establishes that for any two $m \times n$ matrices, their sum is computed by scalar-wise superposition of corresponding entries, yielding the set theoretic union of vector spaces only if operands are identical vectors scaled differently (a specific degenerate case not generally applicable to matrix sets). This operation preserves linearity and serves as a foundational axiom for defining vector space structures over rings or fields where addition forms an abelian group.
Entrywise matrix addition through three worked examples, closing with the commutative and associative properties of matrix addition.