Using the properties of determinant, show that :`|[a^2+1,ab,ac],[ab,b^2+1,bc],[ac,bc,c^2+1]| = 1+a^2+b^2+c^2`

By using properties of determinants, show that : |[1,a,a^2],[1,b,b^2],[1,c,c^2]| = (a-b)(b-c)(c-a)

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

Prove that: |[-a^2, ab,ac],[ba,-b^2,bc],[ca,cb,-c^2]|=4a^2b^2c^2

Using the properties of determinants show that : |[[1, a^2+bc, a^3],[1,b^2+ac,b^3],[1,c^2+ab,c^3]]|=(a-b)(b-c)(c-a)(a^2+b^2+c^2)

Using the properties of determinants show that : |[[1,1,1],[a^2,b^2,c^2],[a^3,b^3,c^3]]|=(a-b)(b-c)(c-a)(ab+bc+ca) .

Prove that: |[a^2,bc,ac+c^2],[a^2+ab,b^2,ac],[ab,b^2+bc,c^2]|=4a^2b^2c^2

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

Using the properties of determinants show that : |[[-bc,b^2+bc,c^2+bc],[a^2+ac,-ac,c^2+ac],[a^2+ab,b^2+ab,-ab]]|=(ab+bc+ca)^3

|[x^2+a^2,ab,ac] , [ab,x^2+b^2,bc] , [ac,bc,x^2+c^2]|=

Using the properties of determinants, show that |(1+a,1,1),(1,1+b,1),(1,1,1+c)|=abc+bc+ca+ab