`" if " |{:(yz-x^(2),,zx-y^(2),,xy-z^(2)),(xz-y^(2),,xy-z^(2),,yz-x^(2)),(xy-z^(2),,yz-x^(2),,zx-y^(2)):}|=|{:(r^(2),,u^(2),,u^(2)),(u^(2),,r^(2),,u^(2)),(u^(2),,u^(2),,r^(2)):}|` then

Options

A. `r^(2)=x+y+z`
B. `r^(2) =x^(2) =y^(2) +z^(2)`
C. `u^(2) =yz+zx+xy`
D. `u^(2) =xyz`

Answer

B, C

Explanation

In the left - hand determinant each element is the cofactor of the elements of the determinant
`|{:(x,,y,,z),(y,,z,,x),(z,,x,,y):}|=Delta " (says) "`
hence
`Delta^(2) = |{:(x,,y,,z),(y,,z,,x),(z,,x,,y):}||{:(x,,y,,z),(y,,z,,x),(z,,x,,y):}|`
`=|{:(x^(2)+y^(2)+z^(2),,xy+yz+zx,,xz+yx+zy),(Sigmaxy,,Sigmax^(2),,Sigmaxy),(Sigmaxy,,Sigmaxy,,Sigmax^(2)):}|`
`|{:(r^(2),,u^(2),,u^(2)),(u^(2),,r^(2),,u^(2)),(u^(2),,u^(2),,r^(2)):}|" "underset(xy+yz+zx=u^(2))("Where "x^(2)+y^(2)+z^(2)=r^(2))`