If `A=[(a,b,c),(b,c,a),(c,a,b)],abc=1,A^(T)A=I,` then find the value of `a^(3)+b^(3)+c^(3).`

Explanation

`therefore" " A^(T)A=I`
`rArr" " |A^(T)A|=|I| rArr |A^(T)||A|=1`
` rArr" " |A||A|=1" "[therefore|A^(T)|=|A|] `
`|A|=pm 1`
` rArr" " |(a,b,c),(b,c,a),(c,a,b)|=pm 1`
` rArr" "3abc-(a^(3)+b^(3)+c^(3))=pm1`
`or" "3-(a^(3)+b^(3)+c^(3))=pm1`
`or" "a^(3)+b^(3)+c^(3)=pm1=2or 4`