Using prperties of determinants, prove that :
`|(1,1,1+3x),(1+3y,1,1),(1,1+3z,1)| = 9(3xyz + xy + yz + zx)`.

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

Using properties of determinants , prove that |(x^2+1,xy,zx),(xy,y^2+1,yz),(zx,yz,z^2+1)|=1+x^2+y^2+z^2

Using properties of determinants, prove that: |[x,x^2,1+px^3],[y,y^2,1+py^3],[z,z^2,1+pz^3]| = (1+pxyz)(x-y)(y-z)(z-x)

Using properties of determinants, prove that : {:|((x+y)^2,zx,zy),(zx,(z+y)^2,xy),(zy,xy,(z+x)^2)|=2xyz(x+y+z)^3

Using the properties of determinants, prove that following : |{:(1,x,x+1),(2x,x(x-1),x(x+1)),(3x(1-x),x(x-1)(x-2),x(x+1)(x-1)):}|=6x^(2)(1-x^(2))

Prove that: {:|(1,x,x^3),(1,y,y^3),(1,z,z^3)| = (x-y)(y-z)(z-x)(x+y+z)

Show that |(1,x^2,x^3),(1,y^2,y^3),(1,z^2,z^3)| = (x-y),(y-z)(z-x)(xy+yz+zx)

Evaluate the following determinants: |(1,3,9),(3,9,1),(9,1,3)|

Show that : |{:(1+x,1,1),(1,1+y,1),(1,1,1+z):}|=xyz(1+(1)/(x)+(1)/(y)+(1)/(z))

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) .