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 |[a, b, c], [b, c, a], [c, a, b]|

Without expanding the determinant,show that (a+b+c) is a factor of the determinant det[[a,b,cb,c,ac,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}

Without expanding the determinant prove that: |0b-c-b0a c-a0|=0

Without expanding determinant show that |(0,a,b),(-a,0,c),(-b,-c,0)|=0

Without expanding, show that the value of each of the determinants is zero: |1/a a^2b c a/bb^2a c a/cc^2a b|

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 the determinant prove that: {:|(0,a,-b),(-a,0,-c),(b,c,0)|:}=0

If a, b, c are positive and unequal, show that value of the determinant Delta=|a b c b c a c a b| is negative.

Using the factor theorem it is found that a+b , b+c and c+a are three factors of the determinant |[-2a ,a+b, a+c],[ b+a,-2b,b+c],[c+a ,c+b,-2c]| . The other factor in the value of the determinant is