Let a, b, c ∈ R be all non - zero and satisfy


Let $a, b, c \in R$ be all non-zero and satisfy $\mathrm{a}^{3}+\mathrm{b}^{3}+\mathrm{c}^{3}=2$. If the matrix

$A=\left(\begin{array}{lll}a & b & c \\ b & c & a \\ c & a & b\end{array}\right)$

satisfies $\mathrm{A}^{\mathrm{T}} \mathrm{A}=\mathrm{I}$, then a value of abc can be :

  1. $\frac{2}{3}$

  2. $-\frac{1}{3}$

  3. 3

  4. $\frac{1}{3}$

Correct Option: , 4


$\mathrm{A}^{\mathrm{T}} \mathrm{A}=\mathrm{I}$

$\Rightarrow a^{2}+b^{2}+c^{2}=1$

and $a b+b c+c a=0$

Now, $(a+b+c)^{2}=1$

$\Rightarrow a+b+c=\pm 1$

So, $a^{3}+b^{3}+c^{3}-3 a b c$

$=(a+b+c)\left(a^{2}+b^{2}+c^{2}-a b-b c-c a\right)$

$=\pm 1(1-0)=\pm 1$

$\Rightarrow 3 \mathrm{abc}=2 \pm 1=3,1$

$\Rightarrow a b c=1, \frac{1}{3}$

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