Умножение выполняется методом "строчка на столбец" . (Дополнительна информация в учебнике: Боревич З.И. Определители и матрицы;![4*A*B*C=4\left[\begin{array}{ccc}0&-18\\21&8\end{array}\right] \left[\begin{array}{ccc}1&-4\\1&2\end{array}\right] = \\ = 4\left[\begin{array}{ccc}0-18&0-18*2\\21+8&-84+16\\\end{array}\right] = 4 \left[\begin{array}{ccc}-18&-36\\29&-68\\\end{array}\right]](https://tex.z-dn.net/?f=4%2AA%2AB%2AC%3D4%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D0%26-18%5C%5C21%268%5Cend%7Barray%7D%5Cright%5D++++%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D1%26-4%5C%5C1%262%5Cend%7Barray%7D%5Cright%5D+%3D+%5C%5C++%3D++4%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D0-18%260-18%2A2%5C%5C21%2B8%26-84%2B16%5C%5C%5Cend%7Barray%7D%5Cright%5D+%3D+4++%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D-18%26-36%5C%5C29%26-68%5C%5C%5Cend%7Barray%7D%5Cright%5D+ "4*A*B*C=4\left[\begin{array}{ccc}0&-18\\21&8\end{array}\right] \left[\begin{array}{ccc}1&-4\\1&2\end{array}\right] = \\ = 4\left[\begin{array}{ccc}0-18&0-18*2\\21+8&-84+16\\\end{array}\right] = 4 \left[\begin{array}{ccc}-18&-36\\29&-68\\\end{array}\right]")
Бурмистрова, Лобанов, Линейная Алгебра (НИУ ВШЭ)
Использовано свойство ассоциативности умножения.
Обращаю внимание, что дистрибутивность относительно умножения матриц невозможна: A*B <> B*A![A= \left[\begin{array}{ccc}2&-2\\3&4\end{array}\right] , B= \left[\begin{array}{ccc}3&-4\\3&5\end{array}\right] , C= \left[\begin{array}{ccc}1&-4\\1&2\end{array}\right] \\ D=2A*(C*2B) =4*A*B*C \\ A*B= \left[\begin{array}{ccc}2*3+2(-3)&-8-10\\9+12&-12+20\end{array}\right] = \left[\begin{array}{ccc}0&-18\\21&8\end{array}\right] \\](https://tex.z-dn.net/?f=A%3D++%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D2%26-2%5C%5C3%264%5Cend%7Barray%7D%5Cright%5D+%2C+B%3D++%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D3%26-4%5C%5C3%265%5Cend%7Barray%7D%5Cright%5D+%2C+C%3D++%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D1%26-4%5C%5C1%262%5Cend%7Barray%7D%5Cright%5D++%5C%5C+D%3D2A%2A%28C%2A2B%29+%3D4%2AA%2AB%2AC+%5C%5C+A%2AB%3D++%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D2%2A3%2B2%28-3%29%26-8-10%5C%5C9%2B12%26-12%2B20%5Cend%7Barray%7D%5Cright%5D+%3D++%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D0%26-18%5C%5C21%268%5Cend%7Barray%7D%5Cright%5D++%5C%5C+ "A= \left[\begin{array}{ccc}2&-2\\3&4\end{array}\right] , B= \left[\begin{array}{ccc}3&-4\\3&5\end{array}\right] , C= \left[\begin{array}{ccc}1&-4\\1&2\end{array}\right] \\ D=2A*(C*2B) =4*A*B*C \\ A*B= \left[\begin{array}{ccc}2*3+2(-3)&-8-10\\9+12&-12+20\end{array}\right] = \left[\begin{array}{ccc}0&-18\\21&8\end{array}\right] \\")