Without expanding, prove that `Delta=|x+y y+z z+x z x y1 1 1|=0`

Without expanding,prove that quad Delta=det[[x+y,y+z,z+xz,x,y1,1,1]]=0

Without expanding prove that det[[x+y,z,1y+z,x,1z+x,y,1]]=0

Show without expanding at any stage that: [x+y,y+z,z+x],[z,x,y],[1,1,1]|=0

Without expanding, show that "Delta"=|(a-x)^2(a-y)^2(a-z)^2(b-x)^2(b-y)^2(b-z)^2(c-x)^2(c-y)^2(c-z)^2|=2(a-b)(b-c)(c-a)(x-y)(y-z)(z-x)

If Delta_(1)=|[a,b, c],[x, y, z],[p,q ,r]|"and "Delta_(2) |[q,-b, y],[-p, a, -x],[r,-c ,z]| then without expanding Delta_(1) " and "Delta_(2), "prove that "Delta_(1) + Delta_(2) =0

Without expanding as far as possible, prove that |{:(1,1,1),(x,y,z),(x^(3),y^(3),z^(3)):}| = (x-y)(y-z)(z-x)(x+y+z) .

Without actual expansion, prove that the following determinants vanish : |{:(x+y,y+z,z+x),(z,x,y),(1,1,1):}|

Without expanding, prove that : |{:(1+b,b+c,c+a),(p+q,q+r,r+p),(x+y,y+z,z+x),(x+y,y+z,z+x):}|=2|{:(a,b,c),(p,q,r),(x,y,z):}|

Using the property of determinants and without expanding in questions 1 to 7 prove that , |{:(x,a,x+a),(y,b,y+b),(z,c,z+c):}|=0

Show without expanding at any stage that: [x,a,x+a],[y,b,y+b],[z,c,z+c]|=0