矩阵乘法的运算规律2.(结合律)(1)(AB)C = A(BC)(2)A(B+C) = AB+ AC（分配律）(B+C)A= BA+CA(3)A.E.=E.Ax=AXI(4)A0 = 0,0A=0a,ba-(5)b84.2矩阵的运算A
§4.2 矩阵的运算 2．矩阵乘法的运算规律 (1) ( ) ( ) AB C A BC = (2) ( ) A B C AB AC + = + (3) A E E A A s n n s s n s n = = ( ) B C A BA CA + = + (5) 1 1 1 1 n n n n a b a b a b a b = (结合律) （分配律) (4) 0 0, 0 0 A A = =
证: 1) 设 A=(a)sn’ B=(bik)nm’ C=(Cu)mr今V= AB=(vik)smW = BC=(W;l)nr)Vix=Zabk, w,-2bucu.其中k=1i-12-2 VC的第i行第1列元素为k=lk=12Zagbacuk=-1 j-1Zagwu=Za,(2bucn)AW的第i行第1列元素为j=1j-1k=12abcu=22bbirCu.结合律得证。aijiCk=1 j=1i=1 k=184.2矩阵的运算
§4.2 矩阵的运算 证：1）设 ( ) , ( ) , ( ) A a B b C c = = = ij sn jk nm kl mr 令 ( ) , ( ) , V AB v W BC w = = = = ik sm jl nr 其中 1 1 , . n m ik ij jk jl jk kl j k v a b w b c = = = = VC 的第 i 行第 l 列元素为 1 1 ( ) m n ij jk kl k j a b c = = = 1 m ik kl k v c = 1 1 m n ij jk kl k j a b c = = = AW 的第 i 行第 l 列元素为 1 n ij jl j a w = 1 1 ( ) n m ij jk kl j k a b c = = = 1 1 n m ij jk kl j k a b c = = = 1 1 . m n ij jk kl k j a b c = = = 结合律得证