[" 20.Using the properties of determinants,prove that "],[[" 1"x+y,x^(2)+y^(2)],[" 1"y+z,y^(2)+z^(2)],[" 1"z+x,z^(2)+x^(2)]]

Using the properties of determinant, show that : |[1,x+y,x^2+y^2],[1,y+z,y^2+z^2],[1,z+x,z^2+x^2]| = (x-y)(y-z)(z-x)

using the properties of determinants prove that |{:(1,x+y,x^2+y^2),(1,y+z,y^2+z^2),(1,z+x,z^2+x^2):}|=(x-y)(y-z)(z-x)

Using properties of determinant prove that: |[1,x+y, x^2+y^2],[1, y+z, y^2+z^2],[1, z+x, z^2+x^2]|= (x-y)(y-z)(z-x)

Using properties of determinant prove that: |[1,x+y, x^2+y^2],[1, y+z, y^2+z^2],[1, z+x, z^2+x^2]|= (x-y)(y-z)(z-x)

Using the properties of determinants, show that : |[[x, y, z],[x^2, y^2, z^2],[x,y,z]]|= 0 .

using properties of determinant prove that {:[( x,x^(2) , 1+ px^(3) ),( y,y^(2) , 1+ py^(2)),( z,z^(2) , 1+pz^(2)) ]:} =( 1+pxyz ) ( x-y) ( y-z ) (z-x) , where p is any scalar .

Using properties of determinants, prove that |(a+x, y, z),(x, a+y, z),(x, y,a+z)|=a^2(a+x+y+z)

Using the properties of determinants, prove that : |[[a+x,y,z],[x,a+y,z],[x,y,a+z]]|=a^2(a+x+y+z)

Using properties of determinants, prove that |[a+x,y,z],[x,a+y,z],[x,y,a+z]|=a^2(a+x+y+z)

Using properties of determinants, prove that |[a+x,y,z],[x,a+y,z],[x,y,a+z]|=a^2(a+x+y+z)