Комплексные числа. Линейная и векторная алгебра. Мозалева Е.М. - 43 стр.

                                         Задание 2
Даны две матрицы          A и B . Найти: A ⋅ B; B ⋅ A.
          3 −1            0        −1 0 2
                                             
      A = 3 5             1 , B =  1 − 8 5 
          4 − 7           5       3  0 2 
                                   

                                   Решение
        Определим тип матриц: A(3×3) , B(3×3) . Число столбцов матрицы A равно
числу строк матрицы B , поэтому произведение A ⋅ B существует, и получим
матрицу типа (3 × 3) , а именно
         3 −1 0  −1 0 2
                              
A ⋅ B =  3 5 1 ⋅  1 − 8 5 =
         4 − 7 5  3      0 2 
                  
   3 ⋅ (− 1) + (− 1) ⋅ 1 + 0 ⋅ 3 3 ⋅ 0 + (− 1) ⋅ (− 8) + 0 ⋅ 0 3 ⋅ 2 + (− 1) ⋅ 5 + 0 ⋅ 2 
                                                                                            
=  3 ⋅ (− 1) + 5 ⋅ 1 + 1 ⋅ 3        3 ⋅ 0 + 5 ⋅ (− 8) + 1 ⋅ 0      3 ⋅ 2 + 5 ⋅ 5 + 1⋅ 2  =
   4 ⋅ (− 1) + (− 7 ) ⋅ 1 + 5 ⋅ 3 4 ⋅ 0 + (− 7 ) ⋅ (− 8) + 5 ⋅ 0 4 ⋅ 2 + (− 7 ) ⋅ 5 + 5 ⋅ 2 
                                                                                            
  − 4     8       1 
                    
=  5 − 40 33 .
   4     56 − 17 
  
        Так как произведение матриц свойством коммутативности не обладает,
то A ⋅ B ≠ B ⋅ A .
        Число столбцов матрицы B равно числу строк матрицы A , поэтому
B ⋅ A существует, и получим матрицу типа (3 × 3) , а именно

          − 1 0 2  3 − 1 0
                                       
B ⋅ A =  1 − 8 5 ⋅  3 5 1 =
          3      0 2   4 − 7 5 
         
   − 1 ⋅ 3 + 0 ⋅ 3 + 2 ⋅ 4 − 1 ⋅ (− 1) + 0 ⋅ 5 + 2 ⋅ (− 7 ) − 1 ⋅ 0 + 0 ⋅ 1 + 2 ⋅ 5 
                                                                                         
= 1 ⋅ 3 + (− 8) ⋅ 3 + 5 ⋅ 4 1 ⋅ (− 1) + (− 8) ⋅ 5 + 5 ⋅ (− 7 ) 1 ⋅ 0 + (− 8) ⋅ 1 + 5 ⋅ 5  =
   3⋅3 + 0⋅3 + 2⋅ 4           3 ⋅ (− 1) + 0 ⋅ 5 + 2 ⋅ (− 7 )     3 ⋅ 0 + 0 ⋅ 1 + 2 ⋅ 5 
  
   5 − 13 10 
               
=  − 1 − 76 17 .
   17 − 17 10 
               
                              − 4  8  1             5 − 13 10 
                                                                
               Ответ: A ⋅ B =  5 − 40 33 ; B ⋅ A =  − 1 − 76 17 .
                               4  56 − 17          17 − 17 10 
                                                                 
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