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

where

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 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 , we have .

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.

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For a 4✕4 series-matrix, *Det A* shall denote the determinant formed of the 4✕4 elements of the matrix. If , then corresponding to *A* there is a *reciprocal* matrix, which we may denote by so that

A matrix