Question

If $A=\left[\begin{array}{ccc}a& b& c\\ b& c& a\\ c& a& b\end{array}\right],abc=1,A^{T}A=I$, then find the value of $a^3+b^3+c^3$.

• A. 1
• B. 2
• C. 3
• D. 4

Answer 1

Answer: (B), (D)

4
2

Given $A=\left[\begin{array}{ccc}a& b& c\\ b& c& a\\ c& a& b\end{array}\right],abc=1,A^{T}A=I$.
$A^{T}A=I$
applying determinant on both sides
$|A{|}^2=1\Rightarrow |A|=\pm 1$
$\Rightarrow |A|=\left|\begin{array}{ccc}a& b& c\\ b& c& a\\ c& a& b\end{array}\right|=\pm 1$
$\Rightarrow (a+b+c)(ab+bc+ca-a^2-b^2-c^2)=\pm 1$
$\Rightarrow a^3+b^3+c^3-3abc=\pm 1$
$\Rightarrow a^3+b^3+c^3=\pm 1+3abc$ where $abc=1$
$\therefore a^3+b^3+c^3=2$ or $4$
Hence, options B and C.