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.!
