Using properties of determinants, prove the following: `|[1,a,a^2],[a^2,1,a],[a,a^2,1]|=(1-a^3)^2`

Using properties of determinants, prove the following: |[1,x,x^2],[x^2, 1,x],[x,x^2,1]|=(1-x^3)^2

Using properties of determinants, prove the following: |[1,x,x^2],[x^2, 1,x],[x,x^2,1]|=(1-x^3)^2

Using properties of determinants,prove the following: det[[1,a,a^(2)a^(2),1,aa,a^(2),1]]=(1-a^(3))^(2)

Using properties of determinants prove the following. abs[[1,x,x^2],[x^2,1,x],[x,x^2,1]]=(1-x^3)^2

Using properties of determinants, prove the following: |[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 the following: |[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 the following: |[a^2 + 1,ab, ac], [ab,b^2 + 1,b c],[ca, cb, c^2+1]|=1+a^2+b^2+c^2

Using properties of determinant, prove that |{:( 1,a,a^(2)),(a^(2) , 1,a),( a,a^(2),1):}|=(a^(3) -1)^(2)

Using properties of determinants, prove the following |(a^2+1,ab,ac),(ab,b^2+1,bc),(ca,cb,c^2+1)|=1+a^2+b^2+c^2 .

By using properties of determinants, show that : |[1,x,x^2],[x^2,1,x],[x,x^2,1]| = (1-x^3)^2