Question

关于线性代数行列式的2道题

-6 -3x -27x^3 81x^4
-6 -3y -27y^3 81y^4
-6 -3z -27z^3 81z^4
-6 -3w -27w^3 81w^4

x y z
设d= y z x
z x y
求-4ax-by -4ay-bz -4az-bx
-4ay-bz -4az-bx -4ax-by
-4az-bx -4ax-by -4ay-bz
第二题结果用d表示

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Answer 1

(1) D = (-39366)*
| 1 x x^3 x^4|
| 1 y y^3 y^4|
| 1 z z^3 z^4|
| 1 w w^3 w^4|
D = (-39366)*
| 1 x x^3 x^4|
| 0 y-x y^3-x^3 y^4-x^4|
| 0 z-x z^3-x^3 z^4-x^4|
| 0 w-x w^3-x^3 w^4-x^4|
D = (-39366)(y-x)(z-x)(w-x)*
| 1 y^2+yx+x^2 y^3+y^2x+yx^2+x^3|
| 1 z^2+zx+x^2 z^3+z^2x+zx^2+x^3|
| 1 w^2+wx+x^2 w^3+w^2x+wx^2+x^3|
D = (-39366)(y-x)(z-x)(w-x)*
| 1 y^2+yx+x^2 y^3+y^2x+yx^2+x^3|
| 0 (z-y)(z+y+x) (z-y)(z^2+y^2+x^2+zy+zx+xy)|
| 0 (w-y)(w+y+x) (w-y)(w^2+y^2+x^2+wy+wx+xy)|
D = (-39366)(y-x)(z-x)(w-x)*(z-y)(w-y)*
[(z+y+x)(w^2+y^2+x^2+wy+wx+xy)
- (w+y+x)(z^2+y^2+x^2+zy+zx+xy)]
D = (-39366)(y-x)(z-x)(w-x)(z-y)(w-y)(w-z)[w(x+y+z)+xy+xz+yz)]

(2) 原式 D = (-4a)*
|x -4ay-bz -4az-bx|
|y -4az-bx -4ax-by|
|z -4ax-by -4ay-bz|
+(-b)*
|y -4ay-bz -4az-bx|
|z -4az-bx -4ax-by|
|x -4ax-by -4ay-bz|
D = (-4a)*
|x -4ay-bz -4az|
|y -4az-bx -4ax|
|z -4ax-by -4ay|
+(-b)*
|y -bz -4az-bx|
|z -bx -4ax-by|
|x -by -4ay-bz|
D = (-4a)^2*
|x -4ay z|
|y -4az x|
|z -4ax y|
+(-b)^2*
|y z -bx|
|z x -by|
|x y -bz|
D = (-4a)^3*
|x y z|
|y z x|
|z x y|
+(-b)^3*
|y z x|
|z x y|
|x y z|
D = (-4a)^3*
|x y z|
|y z x|
|z x y|
+(-b)^3*
|x y z|
|y z x|
|z x y|
D = -d (64a^3+b^3)