Using properties of determinant show that : `|(1+a,1,1),(1,1+b,1),(1,1,1+c)|=abc+ab+bc+ca`

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

Using properties of determinants, prove that |(1,a,a^(3)),(1,b,b^(3)),(1,c,c^(3))| = (a-b)(b-c)(c-a)(a+b+c) .

Using properties of determinant show that : |(a^2,bc,c^2+ac),(a^2+ab,b^2,ac),(ab,b^2+bc,c^2)|=4a^2b^2c^2

Using properties of determinant show that : |(a^2,b^2,c^2),((a+1)^2,(b+1)^2,(c+1)^2),((a-1)^2,(b-1)^2,(c-1)^2)|=4|(a^2,b^2,c^2),(a,b,c),(1,1,1)|

Using properties of determinant show that : |(1+a^2-b^2,2ab,-2b),(2ab,1-a^2+b^2,2a),(2b,-2a,1-a^2-b^2)|=(1+a^2+b^2)^3

Using the properties of determinant, prove that |(a^(2) +1, ab, ac),(ab, b^(2) + 1, bc),(ac, bc, c^(2)+1)| = 1+a^(2) + b^(2) + c^(2) .

Factorise : |(1,a,bc),(1,b,ca),(1,c,ab)|

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

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