# Page:Grundgleichungen (Minkowski).djvu/28

then by AB, the product of the matrices A and B, will be denoted the matrix

${\displaystyle C=\left|{\begin{array}{ccc}c_{11},&\dots &c_{1r}\\\vdots &&\vdots \\c_{p1},&\dots &c_{pr}\end{array}}\right|}$

where

${\displaystyle c_{hk}=a_{h1}b_{1k}+a_{h2}b_{2k}+\dots +a_{hq}b_{qk}\quad \left({h=1,2,\dots p \atop k=1,2,\dots r}\right)}$

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 ${\displaystyle (AB)S=A(BS)}$ 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 ${\displaystyle C=AB}$, we have ${\displaystyle {\bar {C}}={\bar {B}}{\bar {A}}}$.

3°. 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) ${\displaystyle \left|{\begin{array}{cccc}e_{11},&e_{12},&e_{13},&e_{14}\\e_{21},&e_{22},&e_{23},&e_{24}\\e_{31},&e_{32},&e_{33},&e_{34}\\e_{41},&e_{42},&e_{43},&e_{44}\end{array}}\right|=\left|{\begin{array}{cccc}1,&0,&0,&0\\0,&1,&0,&0\\0,&0,&1,&0\\0,&0,&0,&1\end{array}}\right|}$

For a 4✕4 series-matrix, Det A shall denote the determinant formed of the 4✕4 elements of the matrix. If ${\displaystyle DetA\neq 0}$, then corresponding to A there is a reciprocal matrix, which we may denote by ${\displaystyle A^{-1}}$ so that ${\displaystyle A^{-1}A=1}$

A matrix

${\displaystyle f=\left|{\begin{array}{cccc}0,&f_{12},&f_{13},&f_{14}\\f_{21},&0,&f_{23},&f_{24}\\f_{31},&f_{32},&0,&f_{34}\\f_{41},&f_{42},&f_{43},&0\end{array}}\right|}$,