if 3x3 matrix A=[a b c b c a c a b] where a,b,c are positive real numbers such that abc= 1 and A'A=I then find the value of a^3+b^3+c^3

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$$\begin{gathered} \mathrm{Given} \\ \mathrm{A}=\left[\begin{array}{ccc}\mathrm{a}& \mathrm{b}& \mathrm{c}\\ \mathrm{b}& \mathrm{c}& \mathrm{a}\\ \mathrm{c}& \mathrm{a}& \mathrm{b}\end{array}\right] \\ \left[\begin{array}{ccc}\mathrm{a}& \mathrm{b}& \mathrm{c}\\ \mathrm{b}& \mathrm{c}& \mathrm{a}\\ \mathrm{c}& \mathrm{a}& \mathrm{b}\end{array}\right]\times \left[\begin{array}{ccc}\mathrm{a}& \mathrm{b}& \mathrm{c}\\ \mathrm{b}& \mathrm{c}& \mathrm{a}\\ \mathrm{c}& \mathrm{a}& \mathrm{b}\end{array}\right]=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right] \\ \Rightarrow \left[\begin{array}{ccc}{\mathrm{a}}^2+{\mathrm{b}}^2+{\mathrm{c}}^2& \mathrm{ab}+\mathrm{bc}+\mathrm{ca}& \mathrm{ac}+\mathrm{bc}+\mathrm{ab}\\ \mathrm{ab}+\mathrm{bc}+\mathrm{ca}& {\mathrm{a}}^2+{\mathrm{b}}^2+{\mathrm{c}}^2& \mathrm{ac}+\mathrm{bc}+\mathrm{ab}\\ \mathrm{ac}+\mathrm{bc}+\mathrm{ab}& \mathrm{ac}+\mathrm{bc}+\mathrm{ab}& {\mathrm{a}}^2+{\mathrm{b}}^2+{\mathrm{c}}^2\end{array}\right]=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right] \\ \mathrm{so}, \\ {\mathrm{a}}^2+{\mathrm{b}}^2+{\mathrm{c}}^2=1 \\ \mathrm{ab}+\mathrm{bc}+\mathrm{ca}=0 \\ \mathrm{so}, \\ {\left(\mathrm{a}+\mathrm{b}+\mathrm{c}\right)}^2={\mathrm{a}}^2+{\mathrm{b}}^2+{\mathrm{c}}^2+2\left(\mathrm{ab}+\mathrm{bc}+\mathrm{ca}\right)=1+0=1 \\ \mathrm{so}, \\ \mathrm{a}+\mathrm{b}+\mathrm{c}=1 \\ \mathrm{now}, \\ \mathrm{we} \mathrm{know} \mathrm{that}, \\ \left({\mathrm{a}}^3+{\mathrm{b}}^3+{\mathrm{c}}^3-3\mathrm{abc}\right)=\left(\mathrm{a}+\mathrm{b}+\mathrm{c}\right)\left({\mathrm{a}}^2+{\mathrm{b}}^2+{\mathrm{c}}^2-\mathrm{ab}-\mathrm{bc}-\mathrm{ca}\right) \\ \mathrm{given} \mathrm{abc}=1. \\ {\mathrm{a}}^3+{\mathrm{b}}^3+{\mathrm{c}}^3-3=1.\left(0-0\right) \\ {\mathrm{a}}^3+{\mathrm{b}}^3+{\mathrm{c}}^3=3 \end{gathered}$$

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