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1) matrix addition;

2) multiplication by the undelying field elements;

3) matrix multiplication.

Is the last one really obvious? And is it correct? The very definition of algebra implies that any two objects belonging to it can be multiplied. This is certainly not true for matrix multiplication. I see matrix multiplication as abstraction and generalisation of the idea of the inner product (please correct me, if I am wrong). If so, the proper algebraic multiplication operation for matrices would be the Kronecker (direct) product (equivalent and generalisation of the outer product concept). It means, matrix multiplication should be considered as additional structure imposed of matrix algebra that is, strictly speaking, unnecessary for existence for the algebra of matrices, is it?