if x+y+z = 0 then prove that `|{:(x,y,z),(x^2,y^2,z^2),(y+z,z+x,x+y):}|=0`

if x+y+z=0 , then show that, |{:(1,1,1),(x,y,z),(x^3,y^3,z^3):}|=0

|{:(" "0," "x,y),(-x," "0,z),(-y,-z,0):}|=0

If x+y+z=0 then show that abs((1,1,1),(x,y,z),(x^3,y^3,z^3)) = 0.

If a x_1^2+b y_1^2+c z_1^2=a x_2 ^2+b y_2 ^2+c z_2 ^2=a x_3 ^2+b y_3 ^2+c z_3 ^2=d ,a x_2 x_3+b y_2y_3+c z_2z_3=a x_3x_1+b y_3y_1+c z_3z_1=a x_1x_2+b y_1y_2+c z_1z_2=f, then prove that |(x_1, y_1, z_1), (x_2, y_2, z_2), (x_3,y_3,z_3)|=(d-f){((d+2f))/(a b c)}^(1//2)

|{:(x,z,0),(0,y,y),(z,0,x):}|

Without expanding prove that Delta ={:|( x+y,y+z,z+x) ,( z,x,y),( 1,1,1) |:} =0

By using properties of determinants , show that : {:|( x,x^(2) , yz) ,( y,y^(2) , zx ) ,( z , z^(2) , xy ) |:} =( x-y)(y-z) (z-x) (xy+yz+ zx)

If x prop y and y prop z , then prove that (x^7+y^7 +z^7) prop (x^5y^2 +x^3z^4 +y^5z^2)

If z=x+iy and |2z+1|=|z-2i|,then prove that 3(x^2+y^2)+4(x+y)=3

Show that abs((a,b,c),(a+2x,b+2y,c+2z),(x,y,z)) =0