Prove that `|(b+c,a-b,a),(c+a,b-c,b),(a+b,c-a,c)|=3abc-a^(3)-b^(3)-c^(3)`

Show that |(a,b-c,c+b),(a+c,b,c-a),(a-b,b+a,c)|=(a+b+c)(a^(2)+b^(2)+c^(2))

Prove that |(a,b+c,a^(2)),(b,c+a,b^(2)),(c,a+b,c^(2))|=-(a+b+c)xx(a-b)(b-c)(c-a).

Show that |(3a,-a+b,-a+c),(-b+a,3b,-b+c),(-c+a,-c+b,3c)|=3(a+b+c)(ab+bc+ca)

The value of the determinant |(bc,ac,ab),(a,b,c),(a^(3),b^(3),c^(3))| is a)(a + b + c) b)a + b + c -ab c) (a^(2) - b^(2))(b^(2) - c^(2)) (c^(2) - a^(2)) d) a^2+b^2+c^2

Prove that abs[[a-b-c,2a,2a],[2b,b-c-a,2b],[2c,2c,c-a-b]]=(a+b+c)^3

Prove that |[1 , a , a^3],[ 1, b, b^3],[ 1, c, c^3]|=(a-b)(b-c)(c-a)(a+b+c)

Without expanding the determinant, Prove that |[a,a^2, bc],[b,b^2,ca],[c,c^2,ab]|= |[1,a^2,a^3],[1,b^2,b^3],[1,c^2,c^3]|

prove |[1, b c, a(b+c)],[ 1, c a, b(c+a)],[ 1, a b, c(a+b)]|=0