Question

If $A=\left[\begin{array}{ccc}a& b& c\\ b& c& a\\ c& a& b\end{array}\right],abc=1$ and $A^{T}A=I,$ then the value of $a^3+b^3+c^3$ (where $a,b,c>0$ and $A^T$ is transpose of matrix $A$ ) is

• A. $2$
• B. $0$
• C. $1$
• D. $4$

Answer 1

Answer: (D)

$A^{T}A=I$
As $A^T=A$
$\Rightarrow A^2=I$
$\Rightarrow a^2+b^2+c^2=1$ and $ab+bc+ca=0$
As ${\left(a+b+c\right)}^2=a^2+b^2+c^2+2\left(ab+bc+ca\right)$
$\Rightarrow a+b+c=1$
Now, $a^3+b^3+c^3-3abc=\left(a+b+c\right)\left(a^2+b^2+c^2-ab-bc-ca\right)$
$\Rightarrow a^3+b^3+c^3-3=\left(a+b+c\right)$
$\Rightarrow a^3+b^3+c^3=3+1$
$\Rightarrow a^3+b^3+c^3=4$