Question

Let $a, b, c \in R$ be all non-zero and satisfy $a^{3}+b^{3}+c^{3}=2$. If the matrix
$A = \begin{pmatrix}a&b&c\\ b&c&a\\ c&a&b\end{pmatrix}$
satisfies $A ^{ T } A = I ,$ then a value of $abc$ can be

• A. $\frac{2}{3}$
• B. $-\frac{1}{3}$
• C. 3
• D. $\frac{1}{3}$

Answer 1

Answer: (D)

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