B = | _b₁₁_ ......................... _b₁__{_q_} | | | | _b__{_p_ 1} ............. b_{_p_ _q_} | then A + B shall denote the _p_ × _q_ series matrix whose members are _a__{_h_ _k_} + _b__{_h_ _k_}. 2⁰ If we have two matrices 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. For the transposed matrix of C = BA, we have Ċ = ḂĀ 3⁰. We shall have principally to deal with matrices with at most four vertical columns and for horizontal rows. As a unit matrix (in equations they will be known for the sake of shortness as the matrix 1) will be denoted the following matrix (4 × 4 series) with the elements. (34) | e₁₁ e₁₂ e₁₃ e₁₄ | = | 1 0 0 0 | | e₂₁ e₂₂ e₂₃ e₂₄ | | 0 1 0 0 | | e₃₁ e₃₂ e₃₃ e₃₄ | | 0 0 1 0 | | e₄₁ e₄₂ e₄₃ e₄₄ | | 0 0 0 1 | For a 4 × 4 series-matrix, Det A shall denote the determinant formed of the 4 × 4 elements of the matrix. If det A ≠ 0, then corresponding to A there is a reciprocal matrix, which we may denote by A⁻¹ so that A⁻¹A = 1. A matrix _f_ = | 0 _f₁₂_ _f_₁₃ _f₁₄_ | | _f_₂₁ 0 _f₂₃_ _f₂₄_ | | _f₃₁_ _f_₃₂ 0 _f₃₄_ | | _f_₄₁ _f_₄₂ _f_₄₃ 0 | in which the elements fulfil the relation _f__{_h_ _k_} = -_f__{_h_ _k_}, is called an alternating matrix. These relations say that the transposed matrix _ḟ_ = -_f_. Then by _f_^{*} will be the _dual_, alternating matrix (35) _f_^{*} = | 0 _f₃₄_ _f_₄₂ _f₂₃_ | | _f_₄₃ 0 _f₁₄_ _f₃₁_ | | _f₂₄_ _f_₄₁ 0 _f₁₂_ | | _f_₃₂ _f_₁₃ _f_₂₁ 0 |