Find the Rank of `A=[[1,1,1],[a,b,c],[a^2,b^2,c^2]]`

If A=|[1, 1, 1],[a, b, c],[ a^2,b^2,c^2]| , B=|[1,bc, a],[1,ca, b],[1,ab, c]| , then

Without expanding evaluate the determinant "Delta"=|(1, 1, 1),(a, b, c),( a^2,b^2,c^2)| .

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) .

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)

Find 1/2(A+Aprime) and 1/2(A-Aprime) , when A=[[0,a, b],[-a,0,c],[-b,-c,0]]

If the rank of the matrix [[-1,2,5],[2,-4,a-4],[1,-2,a+1]] is 1 then the value of a is (A) -1 (B) 2 (C) -6 (D) 4

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

Without expanding determinant at any stage evaluate |(1,1,1),(a,b,c),(a^(2)-bc,b^(2)-ca,c^(2)-ab)|

Evaluate the following: |[1,,a^2-bc],[1, b,b^2-ac],[1,c,c^2-ab]|

Prove that Delta = |{:(1,,1,,1),(a,,b,,c),(bc+a^(2),,ac+b^(2),,ab+c^(2)):}| = 2(a-b)(b-c)(c-a)