Given a matrix $A =\begin{bmatrix}a&b&c\\b&c&a\\c&a&b\end{bmatrix}$, where a, b, c are real positive numbers. If $abc = 1$ and $A^TA = I$, then find the value of $a^3+b^3+ c^3$.

4

$A^TA = I$

$⇒ |A^TA|=|I|$

$⇒|A|^2=1$

$⇒(a^3+b^3+c^3-3abc)^2=1$

Since a, b, c are positive real number, using AM > GM, we have

$a^3+b^3+c^3≥3abc$

$⇒a^3+b^3+ c^3-3abc = 1$

$⇒a^3+b^3+c^3=4$