3.①)已知A,B为n阶矩阵，且A?+AB=E,r(B)=2,则秩r(AB-2BA-B)= (2)设3阶矩阵A的特征值互不相同，若行列式A=0,则A的秩为 (3)设A是n阶实对称矩阵.满足A1+2A2+42+2A=0.若(A)=r,则A+3= (4已知A是3阶实对称矩阵，满足A+243+42+2A=0,且秩r(4)=2,则4的特征值为一-，秩r(A+ E)= ()设a1=(L,2,-1,0)T,a2=(1,1,0,2)T,ag=(2,1,1,a)T,若由a1,a2,g生成的向量空间的维数 是2，则a= 21-2 (6)设A= 520 ,B是3阶非0矩阵，且AB=0,则a=(). 3a4 2-13 ()设A= 1 ,若存在秩大于1的3阶矩阵B使得BA=0,则An=(). 4(山)设A为3阶矩阵，4=3，A为A的伴随矩阵.交换A的第1行与第2行得矩阵B,则BA1=一 (2)设A为3阶矩阵，A为A的伴随矩阵，14=，则（号A)-1-8A|=( (3).(山)设4阶矩阵A与B相似，B是B的伴随矩阵，若B的特征值是1，-1,2,4，则AA= (②)(2013)设A-(a)为3阶非零矩阵，4为4的行列式，A为a的代数余子式.若a与+A=0,i,j 1.2.3.则1A= 5.()向量a1=(2,1,1,1),a2=(2,1,a,a,g=(3,2,1,a,a4=(4,3,2,1)线性相关且a≠1，则a= (2)已知向量组8=(1，-3,6，-1)7,2=(a,0,6,2)7可以由向量组a1=(1,3,0,5)7,a2=(1,2,1,47,ag= (1,1.2.3)2线性表示则a=.b= =(2,3,a,ag=(，a+2,-27若=(1,3,4可由a1,a2,ag线性表出， 线性表出，则a= (4④向量a-(1,1,1)关于R3的基a1-(1,1,1),02-(0,1,1)T,a3=(0,0,1)F的坐标为--3.(1) ÆA, Bèn› , ÖA2 + AB = E, r(B) = 2, Kùr(AB − 2BA − B2 ) = . (2) 3› AAäpÿÉ”.e1™|A| = 0,KAùè . (3) A¥n¢È°› , ˜vA4 + 2A2 + A2 + 2A = 0, er(A) = r, K|A + 3E| = . (4) ÆA¥3¢È°› ,˜vA4+2A3+A2+2A = 0 ,Öùr(A) = 2, KAAäè ,ùr(A+ E) = . (5) α1 = (1, 2, −1, 0)T , α2 = (1, 1, 0, 2)T , α3 = (2, 1, 1, a) T ,edα1, α2, α3)§ï˛òmëÍ ¥2,Ka = . (6) A =   2 1 −2 5 2 0 3 a 4  ,B¥3ö0› ,ÖAB = 0,Ka = ( ). (7) A =   2 −1 3 a 1 b 4 c 6  , e3ùåu13› B¶BA = 0, KAn = ( ). 4. (1) Aè3› , |A| = 3,A∗èAäë› . ÜA111Ü121› B,K|BA∗ | = . (2) Aè3› ,A∗ èAäë› , |A| = 1 8 ,K|( 1 3A) −1 − 8A∗ | = ( ); (3). (1) 4› AÜBÉq, B∗¥Bäë› , eB∗Aä¥1, −1, 2, 4, K||A|A−1 | = . (2) (2013) A = (aij )è3ö"› ,|A|èA1™, AijèaijìÍ{f™. eaij +Aij = 0, i, j = 1, 2, 3, K|A| = . 5. (1) ï˛α1 = (2, 1, 1, 1), α2 = (2, 1, a, a), α3 = (3, 2, 1, a), α4 = (4, 3, 2, 1)Ç5É'Öa 6= 1, Ka = . (2) Æï˛|β1 = (1, −3, 6, −1)T , β2 = (a, 0, b, 2)Tå±dï˛|α1 = (1, 3, 0, 5)T , α2 = (1, 2, 1, 4)T , α3 = (1, 1, 2, 3)TÇ5L´,Ka = , b = . (3) α1 = (1, 2, 1)T , α2 = (2, 3, a) T , α3 = (1, a + 2, −2)T ,eβ1 = (1, 3, 4)Tådα1, α2, α3Ç5L—,β2 = (0, 1, 2)TÿUdα1, α2, α3Ç5L—,Ka = . (4) ï˛α = (1, 1, 1)'uR3ƒα1 = (1, 1, 1)T , α2 = (0, 1, 1)T , α3 = (0, 0, 1)TãIè . 7