线性代数.几道题,求解用行列式的性质计算下列行列式:(1) | 1 1 1 1 | | -1 1 1 1 | | -1

线性代数.几道题,求解
用行列式的性质计算下列行列式:
(1) | 1 1 1 1 |
| -1 1 1 1 |
| -1 -1 1 1 |
| -1 -1 -1 1 |
(2) | 1 2 3 4 |
| 2 3 4 1 |
| 3 4 1 2 |
| 4 1 2 3 |
(3) | x y x+y |
| y x+y x |
| x+y x y |
用行列式性质证明:

(1) | a1+kb1 b1+c1 c1 | = | a1 b1 c1 |
| a2+kb2 b2+c2 c2 | = | a2 b2 c2 |
| a3+b3 b3+c3 c3 | = | a3 b3 c3 |
(2) | b1+c1 c1+a1 a1+b1 | | a1 b1 c1 |
| b2+c2 c2+a2 a2+b2 | = 2 | a2 b2 c2 |
| b3+c3 c3+a3 a3+b3 | | a3 b3 c3 |
(3) | y+z z+x x+y | | x y z |
| x+y y+z z+x | = 2 | z x y |
| z+y x+y y+z | | y z x |
(4) | a11 a12 0 0 |
| a21 a22 0 0 | = | a11 a12 | | b11 b12 |
| * * b11 b12| | a21 a22 | ·| b21 a22 |
| * * b21 b22|
计算行列式:
(1)
| 1 2 3 ... n-1 n |
| -1 0 3 ... n-1 n |
| -1 -2 0 ... n-1 n |
|... ... ... ... ... ... |
| -1 -2 -3 ... 0 n |
| -1 -2 -3 ... -(n-1) 0 |
(2)
| 1 a1 a2 ... an |
| 1 a1+b1 a2 ... an |
| 1 a1 a2+b2 ... an |
| ... ... ... ... ... |
| 1 a1 a2 ... an+bn |
(3)
| 1+x 1 1 1 |
| 1 1-x 1 1 |
| 1 1 1+y 1 |
| 1 1 1 1-y |
(4)

| x y 0 ... 0 0 |
| 0 x y ... 0 0 |
| ... ... ... ... ... ...|
| 0 0 0 ... x y |
| y 0 0 ... 0 x |
(5)
| x -1 0 ... 0 0 |
| 0 x -1 ... 0 0 |
|... ... ... ... ... ... |
| 0 0 0 ... x -1 |
| an an-1 an-2 ... a2 a1+x|
设 | 1 2 1 2 | | 4 3 2 1 |
A= | 2 1 2 1 |, B= | -2 1 -2 1 |
| 1 2 3 4 | | 0 -1 0 -1 |
求:
(1) 3A-B
(2) 2A+B
(3) 若X满足A+X=B,求X
(4) 若Y满足(2A-Y) +2(B+Y) = 0.求Y
设 | x 0 | | u v | | 3 -4 |
A= | | , B = | | , C = | |
| 7 y | | y 2 | | x v |
且A+2B-C=0,求x, y, u, v的值.
设矩阵A为三阶矩阵,若已知|A|=m , 求 | -mA |.
按分块方法计算下列逆矩阵:
(1) | 1 2 .3 4 |
| 0 1 .2 3 |
| .............|
| 0 0 .1 2 |
| 0 0 .0 1 |
(2) | 1 2 3 .4 |
| 0 1 2 .3 |
| 0 0 1 .2 |
|..............|
| 0 0 0 .1 |
证明:如果对称矩阵A为非奇数矩阵,则A的-1次方也是对称的.
证明: 如果A的平方=A,但A不是单位矩阵，则A必为奇异矩阵.
若n阶矩阵满足A的平方-2A-4I=0 ,试证A+I可逆,并求(A+I)的-1次方.