If `abc=p` and `A=[(a,b,c),(c,a,b),(b,c,a)]`, prove that A is orthogonal if and only if a, b, c are the roots of the equation `x^(3) pm x^(2)-p=0`.

Explanation

Here `A A^(T)=[(a,b,c),(c,a,b),(b,c,a)][(a,c,b),(b,a,c),(c,b,a)]`
`=[(a^(2)+b^(2)+c^(2),ac+ab+bc,ab+bc+ca),(ca+ab+bc,a^(2)+b^(2)+c^(2),cb+ba+ac),(ab+cb+ac,bc+ca+ab,a^(2)+b^(2)+c^(2))]`
Now, `A A^(T)=I` if
`a^(2)+b^(2)+c^(2)=1` and `ab+bc+ca=0`
`implies (a+b+c)^(2)-2(ab+bc+ca)=1` and `ab+bc+ca=0`
`implies a+b+c= pm 1` and `ab+bc+ca=0`
Also, `abc=p`, so that a, b, c are the roots of the equation `x^(3) pm x^(2)-p=0`. Conversely, since a, b, c are the roots of `x^(3) pm x^(2)-p=0, a+b+c= pm 1`, and `ab+bc+ca=0`, we have `a^(2)+b^(2)+c^(2)=1`.