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

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

Show without expanding at any stage that: det [ [a,b,c],[a+2x,b+2y,c+2z],[x,y,z]] =0

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

The value of |[x+y, y+z, z+x] , [z,x,y] , [1,1,1]| is

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

Show that abs([y+z ,x,x],[y,z+x,y],[z,z,x+y])=4xyz

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

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

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) .

Use properties of determinants to evaluate : |{:(x+y,y+z,z+x),(z,x,y),(1,1,1):}|