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