Evaluate the following: ` |[a,b,c],[a^2, b^2, c^2],[a^3, b^3, c^3]| `

Evaluate the following |(1, 1, 1),(a^2,b^2,c^2),(a^3,b^3,c^3)|

Factorize the following |[3,a+b+c,a^3+b^3+c^3],[a+b+c,a^2+b^2+c^2,a^4+b^4+c^4],[a^2+b^2+c^2,a^3+b^3+c^3,a^5+b^5+c^5]|

Factorize the following |3a+b+c a^3+b^3+c^3a+b+c a^2+b^2+c^2a^4+b^4+c^4a^2+b^2+c^2a^3+b^3+c^3a^5+b^5+c^5|

Show that |[a ,b ,c],[ a^2,b^2,c^2],[bc, ca, ab]|=|[1, 1, 1],[a^2,b^2,c^2],[a^3,b^3,c^3]|=(a-b)(b-c)(c-a)(a b+b c+c a) .

If A=|(1,1,1),(a,b,c),(a^3,b^3,c^3)|, B=|(1,1,1),(a^2,b^2,c^2),(a^3,b^3,c^3)|, C=|(a,b,c),(a^2,b^2,c^2),(a^3,b^3,c^3)| , then which relation is correct :

By using properties of determinants. Show that: (i) |[1,a, a^2],[ 1,b,b^2],[ 1,c,c^2]|=(a-b)(b-c)(c-a) (ii) |[1, 1, 1],[a, b, c],[ a^3,b^3,c^3]|=(a-b)(b-c)(c-a)(a+b+c)

Using properties of determinants, prove that following |(a+b+2c,a,b),(c,b+c+2a,b),(c,a,c+a+2b)|=2(a+b+c)^3

If a,b,c are unequal then what is the condition that the value of the following determinant is zero Delta =|(a,a^2,a^3+1),(b,b^2,b^3+1),(c,c^2,c^3+1)|

Prove: |[1,a^2+bc, a^3],[ 1,b^2+c a, b^3],[ 1,c^2+a b, c^3]|=-(a-b)(b-c)(c-a)(a^2+b^2+c^2)

Prove: |(0,b^2a, c^2a),( a^2b,0,c^2b),( a^2c, b^2c,0)|=2a^3b^3c^3