A = | _a₁₁_ ..................... _a__{1 _q_} |
| |
| _a__{_p_ 1} ........... _a__{_p_ _q_} |
B = | _b__{1 1} .............. _b__{1 _r_} |
| |
| _b__{_q_ 1} .......... _b__{_p_ _r_} |
where the number of horizontal rows of B, is equal to the number of
vertical columns of A, then by AB, the product of the matrices A and B,
will be denoted the matrix
C = | _c₁₁_ ...................... _c__{1 _r_} |
| |
| _c__{_p_ _r_} ........... _c__{_p_ _p_} |
where _c__{_h_ _k_} = _a__{_h_ 1} _b₁__{_k_} + _a__{_h_ 2} _b__{2 _h_} +
... _a__{_k_ _s_} _b__{_s_ _k_} + ... + _a__{_k_ _q_} _b__{_q_ _h_}
these elements being formed by combination of the horizontal rows of A
with the vertical columns of B. For such a point, the associative law
(AB)S = A(BS) holds, where S is a third matrix which has got as many
horizontal rows as B (or AB) has got vertical columns.