故A(BC)=(AB)C 2}-(4)验证留做思考题.口 注1乘法的消去律不成立.即 AB=AC+B=C BA=CA+B=C 注2设A为n阶方阵,定义A的幂为4=4A…4显然有 (1)AA=A+ts 2)(47)°=A°; (3)若AB=BA,则(A+B)n=A+CnA-1B+C2A-2B2+…+ Cn-lABn-l+Bn 注30A=0=A0 注4设f(x)=a0+a1x+a2x2+…+ am am,对Anxn,定义f(A):=a0Ln+ a1A+…+ amam. 五.矩阵的转置 定义A=(a1)mxn的转置为n×m矩阵A,其中A的第k行是A的第k 列,1≤k≤n,A'的第r列是A的第r行,1≤r≤m 矩阵的转置满足 1)(A 2)(A+B)=A+B (3)(aA)=aA'; (4)(AB)=BA= Xn r=1 air( X p k=1 brkckj ) = Xn r=1 X p k=1 airbrkckj . s A(BC) = (AB)C. (2)-(4) ➔→➣↔↕➙➛✶ ③ 1 ✲✰✩➜➝✵❻➄➅✶❲ AB = AC ; B = C, BA = CA ; B = C. ③ 2 ❬ A ❭ n ➞ ❈★✮✬✭ A ✩➟❭ Ar := AA · · ·A | {z } r . ➠➡➂ (1) ArAs = Ar+s ; (2) (Ar ) s = Ars; (3) ➢ AB = BA, ➃ (A + B) n = An + C 1 nAn−1B + C 2 nAn−2B2 + · · · + C n−1 n ABn−1 + Bn . ③ 3 0A = 0 = A0. ③ 4 ❬ f(x) = a0 + a1x + a2x 2 + · · · + amx m. ✿ An×n, ✬✭ f(A) := a0In + a1A + · · · + amAm. ➤✶❋●✙➥➦ ✬✭ A = (aij )m×n ✩✸✹❭ n × m ✧★ A0 , ⑤ ♠ A0 ✩❶ k P ♥ A ✩❶ k ◗✮ 1 ≤ k ≤ n, A0 ✩❶ r ◗ ♥ A ✩❶ r P✮ 1 ≤ r ≤ m. ✧★✩✸✹✺✻❫ (1) (A 0 ) 0 = A; (2) (A + B) 0 = A0 + B0 ; (3) (aA) 0 = aA0 ; (4) (AB) 0 = B0A0 . 4