The matrix product: row by column

A rule different from addition

Unlike addition, the product of two matrices is not done cell by cell. To multiply A (of size n x k) by B (of size k x p), the number of columns of A must equal the number of rows of B. The result A x B is then a matrix of size n x p.

The row x column rule

Each coefficient of the result is obtained by taking a whole row of A and a whole column of B, multiplying the terms one by one in order, then adding everything up.

Product A x B, with A = [[1,2],[3,4]] and B = [[5,6],[7,8]]:

         column1   column2
         [ 5 ]     [ 6 ]
         [ 7 ]     [ 8 ]

row1 [1 2]  ->  1*5 + 2*7 = 5+14 = 19   |   1*6 + 2*8 = 6+16 = 22
row2 [3 4]  ->  3*5 + 4*7 = 15+28 = 43  |   3*6 + 4*8 = 18+32 = 50

Result:
         [ 19  22 ]
A x B =  [ 43  50 ]

(coefficient at row 1 column 1 = term-by-term product of row 1 of A and column 1 of B, then sum)

General method

To find the coefficient at position (i, j) of the product: take the whole row i of A, the whole column j of B, multiply each pair of aligned terms, then add all these products. Each coefficient of the result therefore requires a sum of products, not a simple multiplication.

Common pitfall

The matrix product is in general NOT commutative: A x B is almost always different from B x A (and B x A may not even exist if the sizes do not match in that order). You must also check that the sizes are compatible (columns of A = rows of B) before even starting the calculation, otherwise the product does not exist.