will be known as the transposed matrix of A, and will be denoted by Ā.
Ā = | a_{1 1} . . . . . . . . . . . . . a_{p 1} |
| |
| a_{1 q} . . . . . . . . . . . . a_{p q} |
If we have a second p × q series matrix B,
B = | b_{1 1} . . . . . . . . . . . . . . . . . b_{1 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}.
2^o If we have two matrices
A= | a_{1 1} . . . . . . . . . . . . a_{1 q} | B = | b_{1 1} . . . . . . . . . . b_{1 r} |
| | | |
| a_{p 1} . . . . . . . . . . . a_{p q} | | 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 matrics A and B, will be denoted the matrix
C = | c_{1 1} . . . . . . . . . . . c_{1 r} |
| |
| c_{p r} . . . . . . . . . . . c_{p p} |
where c_{h k} = a_{h 1} b_{1 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 C̄ = B̄Ā
