If `x^3=1`, prove that `Det [[a,b,c],[b,c,a],[c,a,b]]`=`(a+bx+cx^2)Det[[1,b,c],[x^2,c,a],[x,a,b]]`

prove that det[[1,a,b+c1,b,c+a1,c,a+b]]=0

det[[b+c,a,bc+a,c,aa+b,b,c]]

Show that det[[1,a,b_(-)c1,b,c+a1,c,a+b]]=0

if x^(3) =1,prove : |{:(a,b,c),(b,c,a),(c,a,b):}|=(a+bx+cx^2)|{:(1,b,c),(x^2,c,a),(x,a,b):}|

Prove that det[[a-b,b-c,c-ab-c,c-a,a-bc-a,a-b,b-c]]=0

det[[1,a,a+b1,b,b+c1,c,c+a]]=0

det[[b+c,a,bc+a,c,aa+b,b,c]]=(a+b+c)(a-c)^(2)

Prove that det[[1,a,a^(2)-bc1,b,b^(2)-ca1,c,c^(2)-ab]]=0

Prove that det[[a,b+c,a^(2)b,c+a,b^(2)c,a+b,c^(2)]]=-(a+b+c)xx(a-b)(b-c)(c-a)

Using properties of determinant show that: det[[1,a,-bc1,b,-ca1,c,-ab]]=(a-b)(b-c)(c-a)det[[1,b,-ca1,c,-ab]]=(a-b)(b-c)(c-a)