" 8.(i) "|[1,a,a_(1)^(2)],[1,b,b^(2)],[1,c,c^(2)]|=(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)

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)

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)

|(1,a^(2),a^(4)),(1,b^(2),b^(4)),(1,c^(2),c^(4))|=(a+b)(b+c)(c+a)|(1,1,1),(a,b,c),(a^(2),b^(2),c^(2))|

If {[([3,1,2],[8,9,5],[1,1,3])([1,3,3],[3,2,7],[3,7,9])([3,8,1],[1,9,1],[2,5,3])])}^(2)=([a_(1),a_(2),a_(3)],[b_(1),b_(2),b_(3)],[c_(1),c_(2),c_(3)]) then the value of |a_(2)-b_(1)|+|a_(3)-c_(1)|+|b_(3)-c_(2)| is

Show that |[a_(1),b_(1),-c_(1)],[-a_(2),-b_(2),c_(2)],[a_(3),b_(3),-c_(3)]|=|[a_(1),b_(1),c_(1)],[a_(2),b_(2),c_(2)],[a_(3),b_(3),c_(3)]|

Prove: |(1,a, b c),(1,b ,c a),(1,c ,a b)|=|(1,a ,a^2),( 1,b,b^2),( 1,c,c^2)|

Prove that =|1 1 1a b c b c+a^2a c+b^2a b+c^2|=2(a-b)(b-c)(c-a)

Prove that =|1 1 1a b c b c+a^2a c+b^2a b+c^2|=2(a-b)(b-c)(c-a)

If A_(i),B_(i),C_(i) are the cofactors of a_(i),b_(i),c_(i) respectively,i=1,2,3 in Delta =|(a_(1),b_(2),c_(1)),(a_(2),b_(2),c_(2)),(a_(3),b_(3),c_(3))| show that |(A_(1),B_(1),C_(1)),(A_(2),B_(2),C_(2)),(A_(3),B_(3),C_(3))|=Delta^(0)