Question

suppose $a,b,c\in R$ and $abc=1$. If $A=\left[\begin{array}{ccc}2a& b& c\\ b& 2c& a\\ c& a& 2b\end{array}\right]$ is such that $A^{\prime }A=4^{1/3}I$ and $|A|>0$, find the value of $a^3+b^3+c^3$.

Answer 1

$Givenabc=1andA=\left[\begin{array}{ccc}2a& b& c\\ b& 2c& a\\ c& a& 2b\end{array}\right]suchthat{A}^{T}A=4^{1/3}Iand|A|>0$

$A^{T}A=4^{\frac{1}{3}}I\Rightarrow \left|A^{T}A\right|={\left(4^{\frac{1}{3}}\right)}^3\left|I\right|$

$\Rightarrow {\left|A\right|}^2=4\Rightarrow \left|A\right|=2$

$\left|A\right|=\left|\begin{array}{ccc}2a& b& c\\ b& 2c& a\\ c& a& 2b\end{array}\right|=10abc-2(a^3+b^3+c^3)=2$

$Hence,a^3+b^3+c^3=4$