Аналитическая геометрия и линейная алгебра. I семестр. Курс лекций. Кирсанов А.А. - 22 стр.

22
Ïðèìåð.
                        5 4         1 0 3
                  A =      , B =       .
                       6 7          3 1 2
    Òàê êàê ÷èñëî ñòîëáöîâ ìàòðèöû A ðàâíî ÷èñëó ñòðîê ìàò-
ðèöû B , ìû ìîæåì ñîñòàâèòü èõ ïðîèçâåäåíèå:

              5 4   1 0 3   5 ⋅1 + 4 ⋅ 3 5 ⋅ 0 + 4 ⋅ 1 5 ⋅ 3 + 4 ⋅ 2 
C = A ⋅ B =       ⋅     =                                        =
              6 7   3 1 2   6 ⋅1 + 7 ⋅ 3 6 ⋅ 0 + 7 ⋅1 6 ⋅ 3 + 7 ⋅ 2 
                                    17 4 23 
                                =           .
                                    27 7 32 
    Çàìåòèì, ÷òî ïðîèçâåäåíèå B ⋅ A ñîñòàâèòü íåëüçÿ.
    Ïðèâåä¸ííûé ïðèìåð ïîêàçûâàåò, ÷òî äëÿ ìàòðèö íå âûïîë-
íÿåòñÿ êîììóòàòèâíûé çàêîí óìíîæåíèÿ, ò.å. â îáùåì ñëó÷àå
                          A⋅ B ≠ B ⋅ A .
    Åñëè èìååò ìåñòî ðàâåíñòâî
                          A⋅ B = B ⋅ A ,
ìû áóäåì ãîâîðèòü, ÷òî ìàòðèöû A è B êîììóòèðóþò èëè ïåðå-
ñòàíîâî÷íû. Íåîáõîäèìûì óñëîâèåì ïåðåñòàíîâî÷íîñòè ìàòðèö
ÿâëÿåòñÿ èõ ïðèíàäëåæíîñòü ê ìíîæåñòâó êâàäðàòíûõ ìàòðèö,
îäíàêî ýòî óñëîâèå íå ÿâëÿåòñÿ äîñòàòî÷íûì.
Ïðèìåð.
               1 2          −1 8                − 9 22 
          A =      , B =       . AB = BA =          .
               −1 3         − 4 7               − 11 13 

     Èç ôîðìóëû (1.9) ñ î÷åâèäíîñòüþ ñëåäóþò ñëåäóþùèå ñâîé-
ñòâà ïðîèçâåäåíèÿ ìàòðèö:
     1. Ñî÷åòàòåëüíîå
                              (AB )C = A(BC ) ;
    2. Ðàñïðåäåëèòåëüíîå ñâîéñòâî óìíîæåíèÿ ìàòðèö îòíîñè-
òåëüíî ñëîæåíèÿ
                            (A + B )C = AC + BC ,