Without expanding show that : `|(a+b,b+c,c+a),(b+c,c+a,a+b),(c+a,a+b,b+c)|=2|(a,b,c),(b,c,a),(c,a,b)|`

Without expanding show that : |(a,b+c,1),(b,c+a,1),(c,a+b,1)|=0

Without expanding show that : |(1,bc,bc(b+c)),(1,ca,ca(c+a)),(1,ab,ab(a+b))|=0

Without expanding show that : |(1,a,a^2),(1,b,b^2),(1,c,c^2)|=|(1,bc,b+c),(1,ca,c+a),(1,ab,a+b)|

Without expanding show that : |(1,a,a^2-bc),(1,b,b^2-ca),(1,c,c^2-ab)|=0

Without expanding show that : |(bc,a,a^2),(ca,b,b^2),(ab,c,c^2)|=|(1,a^2,a^3),(1,b^2,b^3),(1,c^2,c^3)|

Using properties of determinant show that : |(a,b+c,a^2),(b,c+a,b^2),(c,a+b,c^2)|=-(a+b+c)(a-b)(b-c)(c-a)

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

Using properties of determinant show that : |(a,b,c),(a^2,b^2,c^2),(b+c,c+a,a+b)|=(a+b+c)(a-b)(b-c)(c-a)

Show that a + b + c = c + a + b

Find the value of : |(b+c,a,a),(b,c+a,b),(c,c,a+b)|