线性代数，计算行列式
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不是范德蒙行列式。后 3 列分别减去第 1 列， 得 D =
|1 0 0 0|
|a1 a2-a1 a3-a1 a4-a1|
|(a1)^2 (a2)^2-(a1)^2 (a3)^2-(a1)^2 (a4)^2-(a1)^2|
|(a1)^4 (a2)^4-(a1)^4 (a3)^4-(a1)^4 (a4)^4-(a1)^4|
D =
|a2-a1 a3-a1 a4-a1|
|(a2)^2-(a1)^2 (a3)^2-(a1)^2 (a4)^2-(a1)^2|
|(a2)^4-(a1)^4 (a3)^4-(a1)^4 (a4)^4-(a1)^4|
D = (a2-a1)(a3-a1)(a4-a1)*
|1 1 1|
|a2+a1 a3+a1 a4+a1|
|[(a2)^2+(a1)^2](a2+a1) [(a3)^2+(a1)^2](a3+a1) [(a4)^2+(a1)^2](a4+a1)|
D = (a2-a1)(a3-a1)(a4-a1)*
|1 0 0|
|a2+a1 a3-a2 a4-a2|
|[(a2)^2+(a1)^2](a2+a1) (a3-a2)x (a4-a2)y|
其中：x = (a1)^2+(a2)^2+(a3)^2+a1a2+a1a3+a2a3
y = (a1)^2+(a2)^2+(a4)^2+a1a2+a1a4+a2a4
则 D = (a2-a1)(a3-a1)(a4-a1)(a3-a2)(a4-a2)(y-x)
= (a2-a1)(a3-a1)(a4-a1)(a3-a2)(a4-a2)(a4-a3)(a1+a2+a3+a4)
|1 0 0 0|
|a1 a2-a1 a3-a1 a4-a1|
|(a1)^2 (a2)^2-(a1)^2 (a3)^2-(a1)^2 (a4)^2-(a1)^2|
|(a1)^4 (a2)^4-(a1)^4 (a3)^4-(a1)^4 (a4)^4-(a1)^4|
D =
|a2-a1 a3-a1 a4-a1|
|(a2)^2-(a1)^2 (a3)^2-(a1)^2 (a4)^2-(a1)^2|
|(a2)^4-(a1)^4 (a3)^4-(a1)^4 (a4)^4-(a1)^4|
D = (a2-a1)(a3-a1)(a4-a1)*
|1 1 1|
|a2+a1 a3+a1 a4+a1|
|[(a2)^2+(a1)^2](a2+a1) [(a3)^2+(a1)^2](a3+a1) [(a4)^2+(a1)^2](a4+a1)|
D = (a2-a1)(a3-a1)(a4-a1)*
|1 0 0|
|a2+a1 a3-a2 a4-a2|
|[(a2)^2+(a1)^2](a2+a1) (a3-a2)x (a4-a2)y|
其中：x = (a1)^2+(a2)^2+(a3)^2+a1a2+a1a3+a2a3
y = (a1)^2+(a2)^2+(a4)^2+a1a2+a1a4+a2a4
则 D = (a2-a1)(a3-a1)(a4-a1)(a3-a2)(a4-a2)(y-x)
= (a2-a1)(a3-a1)(a4-a1)(a3-a2)(a4-a2)(a4-a3)(a1+a2+a3+a4)
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