Matrix multiplication begins with the column-wise definition: AB is formed by applying A to each column of B. From that definition, the entry-wise row-column rule follows. The same product can then be reorganized row-wise, as a sum of outer products, in blocks and through special products such as diagonal, shear, orthonormal and upper triangular matrices.
Matrix multiplication is one operation with several equivalent readings. Each reading is useful for a different kind of reasoning.
The chapter moves from definitions to reorganizations, then to special products where the structure of the factors controls the structure of the result.
Each entry cij of C = AB is the dot product of row i of A with column j of B. This entry-wise rule is not a separate definition: it follows directly from the column-wise meaning of matrix multiplication.
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It shows that a matrix product can be built by adding simple rank-1 layers. Each layer reads one scalar from the input and places the result in one column direction of A. This viewpoint leads naturally to low-rank approximation and later ideas such as SVD.