线性代数︳ax+byay+byaz+bx︳︳xyz︳︳ay+bzaz+bxax+by︳=(a^3+b^3)︳yzx︳︳az+bxax+byay+bz︳︳zxy︳

Question

线性代数
︳ax+by ay+by az+bx︳ ︳x y z︳
︳ay+bz az+bx ax+by︳=(a^3+b^3)︳y z x ︳
︳az+bx ax+by ay+bz ︳ ︳z x y ︳

Answer 1

这个行列式要用行列式的分拆性质计算
注意行列式的分拆方法,应该分拆成 2*2*2 = 8 个行列式的和
但其中6个行列式由于有两列成比例为0,只有2个行列式不为0.
左行列式 =
ax ay+bz az+bx by ay+bz az+bx
ay az+bx ax+by + bz az+bx ax+by
az ax+by ay+bz bx ax+by ay+bz
=
ax ay az+bx ax bz az+bx by ay az+bx by bz az+bx
ay az ax+by + ay bx ax+by + bz az ax+by + bz bx ax+by
az ax ay+bz az by ay+bz bx ax ay+bz bx by ay+bz
(第3个行列式的1,2列成比例,故等于0)
=
ax ay az ax ay bx ax bz az ax bz bx by bz az by bz bx
ay az ax + ay az by + ay bx ax + ay bx by + bz bx ax + bz bx by
az ax ay az ax bz az by ay az by bz bx by ay bx by bz
(2,3,4,5个行列式:1,3列,2,3列,1,3列,2,3列成比例,故等于0)
=
ax ay az by bz bx
ay az ax + bz bx by
az ax ay bx by bz
提取公因子,第2个行列式第3列依次与第2,1列交换,得
= x y z
(a^3+b^3) y z x
z x y
= 右行列式.