线性方程组的矩阵形式Ax=b 线性变换的矩阵形式y=Ax 3.方阵的幂: A,k,l为正整数 A=A,A4+=AA(k=12,… 算律:(1)A4A=A (2)(A4) 101 例4A=20,求A(k=23,…) 01T10 102 解法1A2=2020 02T1011「103 A=A2A 22020 230 可以验证:A 101 001 解法2A=20=2+000=B+C BC=CB→(B+C)=B+kBC+…+Ck C2=O→4=(B+C)=B+kBC +k2 1‖10005 线性方程组的矩阵形式 Ax = b 线性变换的矩阵形式 y = Ax 3. 方阵的幂： Ann , k , l 为正整数 A = A 1 , ( 1,2, ) A k+1 = A k A k =  算律：(1) k l k l A A A + = (2) k l k l (A ) = A 例 4           = 1 2 0 1 0 1 A , 求 A (k = 2,3, ) k ． 解法 1           =                     = 1 2 0 1 0 2 1 2 0 1 0 1 1 2 0 1 0 1 2 2 A           =                     = = 1 2 0 1 0 3 1 2 0 1 0 1 1 2 0 1 0 2 3 2 2 3 A A A 可以验证：           = 1 2 0 1 0 k k k A 解法 2 A = B + C           +           =           = 0 0 0 0 0 0 0 0 1 1 2 1 1 2 0 1 0 1 BC = CB k k k k B +C = B + kB C + +C ( ) −1  C 2 = O A B C B kB C k k k k 1 ( ) − = + = +                     +           = − 0 0 0 0 0 0 0 0 1 1 2 1 1 2 1 k k 1 k