作辅助行列式D1=
11111
abcdx
a^2b^2c^2d^2x^2
a^3b^3c^3d^3x^3
a^4b^4c^4d^4x^4
此为Vandermonde行列式,故
D1=(b-a)(c-a)(d-a)(c-b)(d-b)(d-c)(x-a)(x-b)(x-c)(x-d).
又因为行列式D1中x^3的系数-M45即为行列式D
所以
D=-(b-a)(c-a)(d-a)(c-b)(d-b)(d-c)(-a-b-c-d)
=(a-b)(a-c)(a-d)(b-c)(b-d)(c-d)(a+b+c+d).