Find the values of `x ,\ y,\ z` if the matrix
`A=[(0,2y,z), (x, y,-z ),(x,-y, z)]` satisfy the equation `A^T\ A=I_3` .

Explanation

It is given that `A=[(0,2y,z),(x,y,-z),(x,-y,z)]`
`:. A^(T)=[(0,x,x),(2y,y,-y),(z,-z,z)]`
Now, `A^(T)A=I`
`implies [(0,x,x),(2y,y,-y),(z,-z,z)][(0,2y,z),(x,y,-z),(x,-y,z)]=[(1,0,0),(0,1,0),(0,0,1)]`
`[(0+x^(2)+x^(2),0+xy-xy,0-xz+xz),(0+xy-xy,4y^(2)+y^(2)+y^(2),2yz-yz-yz),(0-xz+zx,2yz-yz-yz,z^(2)+z^(2)+z^(2))]`
or `[(2x^(2),0,0),(0,6y^(2),0),(0,0,3z^(2))]=[(1,0,0),(0,1,0),(0,0,1)]`
On comparing the corresponding elements, we have
`2x^(2)=1` or `x= pm 1/sqrt(2)`
`6y^(2)=1` or `y= pm 1/sqrt(6)`
`3z^(2)=1` or `z = pm 1/sqrt(3)`