If $a b c = 1$ and $A =$
a b c
c a b
b c a
is an orthogonal matrix then
COLUMN-I
COLUMN-II
A) The value of $a + b + c$ can be P) $- 1$
B) The value of $a b + b c + c a$ is Q) $0$
C) The value of $a^{2} + b^{2} + c^{2}$ is R) $1$
D) The value of $a^{3} + b^{3} + c^{3}$ S) $2$

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Solution

$A =$
a b c
c a b
b c a

is orthogonal matrix
$A A^{T} = I = A^{T} A$
$a_{1}$ $a_{2}$ $a_{3}$
$b_{1}$ $b_{2}$ $b_{3}$
$c_{1}$
$c_{2}$
$c_{3}$

is orthogonal
and if
$a = a_{1}^i + a_{2}^j + a_{3}^k$
$b = b_{1}^i + b_{2}^j + b_{3}^k$
$c = c_{1}^i + c_{2}^j + c_{3}^k$
then $| a | = 1$
$| b | = 1$
$| c | = 1$
$a . b = b . c = c . a = 0$
hence
$a_{1} = b_{2} = c_{3} = a$
$a_{2} = b_{3} = c_{1} = b$
$a_{3} = b_{1} = c_{2} = c$
$a^{2} + b^{2} + c^{2} = 1$ and
$a b + b c + a c = 1$

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