If matrix $A=\left[\begin{array}{lll}a & b & c \ b & c & a \ c & a & b\end{array} ight]$ where $a

Question

If matrix $A=\left[\begin{array}{lll}a & b & c \ b & c & a \ c & a & b\end{array}ight]$ where $a, b, c$ are real positive numbers, $a b c=1$ and $A^{T} A=I$, then find the value of $a^{3}+b^{3}+c^{3}$.

Professor PC Thomas Classes · Answer

$A^{T} A=I$\ $\left[\begin{array}{lll}a & b & c \ b & c & a \ c & a & b\end{array}ight]\left[\begin{array}{lll}a & b & c \ b & c & a \ c & a & b\end{array}ight]=\left[\begin{array}{lll}1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1\end{array}ight]$\ $\Rightarrow\left[\begin{array}{lll}a^{2}+b^{2}+c^{2} & a b+b c+c a & a b+b c+c a \ a b+b c+c a & a^{2}+b^{2}+c^{2} & a b+b c+c a \ a b+b c+c a & a b+b c+c a & a^{2}+b^{2}+c^{2}\end{array}ight]$ $=\left[\begin{array}{lll}1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1\end{array}ight]$ $$\begin{equation*} \Rightarrow \quad a^{2}+b^{2}+c^{2}=1 	ag{1} \end{equation*}$$ and $a b+b c+c a=0$\ Now $a^{3}+b^{3}+c^{3}$ $$\begin{equation*} =(a+b+c)\left(a^{2}+b^{2}+c^{2}-a b-b c-c aight)+3 a b c 	ag{2} \end{equation*}$$$$\begin{equation*} =(a+b+c)+3 	ag{3} \end{equation*}$$ Now $(a+b+c)^{2}=a^{2}+b^{2}+c^{2}+2(a b+b c+c a)$ $$ =1+2 \cdot 0=1 $$ $\Rightarrow \quad a+b+c=1$\ (since $a, b, c$ are real positive numbers)\ Now from (3), $a^{3}+b^{3}+c^{3}=1+3=4$\