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 $a^3+b^3+ c^3$.
Answer & explanation
Correct answer: option 3
$A^TA = I$
$⇒ |A^TA|=|I|$
$⇒|A|^2=1$
$⇒(a^3+b^3+c^3-3abc)^2=1$
Since a, b, c are positive real number, using AM > GM, we have
$a^3+b^3+c^3≥3abc$
$⇒a^3+b^3+ c^3-3abc = 1$
$⇒a^3+b^3+c^3=4$