Without expanding, show that the value of each of the determinants is zero: `|1/a a^2 bc 1/b b^2 ac 1/c c^2 ab|`

Without expanding, show that the value of each of the determinants is zero: |[1,a, a^2-bc],[1,b,b^2-ac],[1,c,c^2-ab]|

Without expanding, show that the value of each of the following determinants is zero: |1//a , a^2b c1//bb^2a c1//cc^2a b| (ii) |a+b2a+b3a+b2a+b3a+b4a+b4a+b5a+b6a+b| (iii) |1a a^2 1bb^2 1cc^2\ \ \ -b c-a c-a b|

Without expanding, show that the value of each of the determinants is zero: |a+b2a+b3a+b2a+b3a+b4a+b4a+b5a+b6a+b|

Without expanding, show that the value of the following determinant is zero: |[49, 1, 6], [39, 7, 4], [26, 2, 3]|

Using properties of determinants, show that |1 a a^2 -b c 1 b b^2 -c a 1 c c^2 -a b|=0

Without expanding the determinant, show that (a+b+c) is a factor of the determinant |[a, b, c], [b, c, a], [c, a, b]|

Without expanding the determinant, show that (a+b+c) is a factor of the determinant of . \begin{pmatrix} a & b & c \\ b & c & a \\ c & a & b \\ \end{pmatrix}

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)|

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

Prove that the value of each the following determinants is zero: |[(a^x+a^(-x))^2,(a^x-a^(-x))^2 ,1],[(b^y+b^(-y))^2,(b^y-b^(-y))^2 ,1],[(c^z+c^(-z))^2,(c^z-c^(-z))^2, 1]|