Given a matrix `A=[(a,b,c), (b,c,a), (c,a,b)],where a ,b ,c` are real positive numbers `a b c=1a n dA^T A=I ,` then find the value of `a^3+b^3+c^3dot`

Explanation

`A^(T)A=I`
`implies |A^(T)A|=|I|`
`implies |A|^(2)=1`
`implies (a^(3)+b^(3)+c^(3) -3abc)^(2) =1`
Since, a, b, c are positive real number, using `AM ge GM`, we have
`a^(3)+b^(3)+c^(3) ge 3abc`
`implies a^(3)+b^(3)+c^(3)-3abc=1`
`implies a^(3)+b^(3)+c^(3)=4`