求解答行列式a0a0a；b0c0d；b^20c^20d^2；0ab0bc0;0cd0ad0.a0a0ab0c0db^20c^20d^20ab0bc00cd0ad0

Question

求解答行列式a 0 a 0 a； b 0 c 0 d；b^2 0 c^2 0 d^2； 0 ab 0 bc 0; 0 cd 0 ad 0.
a 0 a 0 a
b 0 c 0 d
b^2 0 c^2 0 d^2
0 ab 0 bc 0
0 cd 0 ad 0

Answer 1

按第二列展开：
D=[(-1)^(4+2)]*ab*|(a,a,0,a)(b,c,0,d)(b^2,c^2,0,d^2)(0,0,ad,0)|
+[(-1)^(5+2)]*cd*|(a,a,0,a)(b,c,0,d)(b^2,c^2,0,d^2)(0,0,bc,0)|
两行列式再第四行展开：
=-a^2bd*|(a,a,a)(b,c,d)(b^2,c^2,d^2)|+c^2bd|(a,a,a)(b,c,d)(b^2,c^2,d^2)|
=bda(c^2-a^2)(d-c)(d-b)(c-b)
=abd(a+c)(a-c)(b-c)(b-d)(c-d)