The reqd. Determinant (Det.) D is given by,
D=|(1,a,a^2,a^3),(1,b,b^2,b^3),(1,c,c^2,c^3),(1,d,d^2,d^3)|.
Then, applying R_2-R_1, R_3-R_1,R_4-R_1 and expanding
the resulting det. by C_1, we get,
D=|(b-a,b^2-a^2,b^3-a^3),(c-a,c^2-a^2,c^3-a^3),(d-a,d^2-a^2,d^3-a^3)|,
=a(b-a)(c-a)(d-a)|(1,b+a,b^2+ba+a^2),(1,c+a,c^2+ca+a^2),(1,d+a,d^2+da+a^2)|.
Next, we apply R_2-R_1, R_3-R_1, to get, D=kD_1, where,
k=a(b-a)(c-a)(d-a), and,
D_1,
=|(1,b+a,b^2+ba+a^2),(0,c-b,c-b*c+b+a),(0,d-b,d-b*d+b+a)|,
=(c-b)(d-b)|(1,b+a,b^2+ba+a^2),(0,1,c+b+a),(0,1,d+b+a)|,
Expanding D_1 by C_1, we get,
D_1=(c-b)(d-b)|(1,c+b+a),(1,d+b+a)|,
=(c-b)(d-b)(d-c).
rArrD=a(b-a)(c-a)(d-a)(c-b)(d-b)(d-c).
Enjoy Maths.!
