A matrix algebra is an algebra (see More about Algebras) the elements of which are matrices.
There is a canonical isomorphism of a matrix algebra onto a row space (see chapter Row Spaces) that maps a matrix to the concatenation of its rows. This makes all computations with matrix algebras that use its vector space structure as efficient as the corresponding computation with a row space. For example the computation of a vector space basis, of coefficients with respect to such a basis, and of representatives under the action on a vector space by right multiplication.
If one is interested in matrix algebras as domains themselves then one should think of this algebra as of a row space that admits a multiplication. For example, the convention for row spaces that the coefficients field must contain the field of the vector elements also applies to matrix algebras. And the concept of vector space bases is the same as that for row spaces (see Bases for Matrix Algebras).
In the chapter about modules (see chapter Modules) it is stated that
modules are of interest mainly as operation domains of algebras. Here
we can state that matrix algebras are of interest mainly because they
describe modules. For some of the functions it is not obvious whether
they are functions for modules or for algebras or for the matrices that
generate an algebra. For example, one usually talks about the
fingerprint of an A-module M, but this is in fact computed as the
list of nullspace dimensions of generators of a certain matrix algebra,
namely the induced action of A on M as is computed using
Operation( A, M ) (see Fingerprint, Operation for Algebras).
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