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If $a, b, c \in R, A = [\begin{matrix} 0 & 2 b & c \\ a & b & - c \\ a & - b & c \end{matrix}]$ is an orthogonal matrix, then match the following:
Column-I Column-II
(I) $a^{2} + b^{2} =$ (P) $1$
(II) $b^{2} - c^{2} =$ (Q) $\frac{5}{6}$
(III) $c^{2} + a^{2} =$ (R) $- \frac{1}{6}$
(IV) $a^{2} + b^{2} + c^{2} =$ (S) $\frac{2}{3}$

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$(I) - (Q); (I I) - (P); (I I I) - (S); (I V) - (R)$

$(I) - (Q); (I I) - (R); (I I I) - (S); (I V) - (P)$

$(I) - (S); (I I) - (P); (I I I) - (Q); (I V) - (R)$

$(I) - (S); (I I) - (R); (I I I) - (Q); (I V) - (P)$

answer is A.

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Detailed Solution

Equate Corresponding elements of $A A^{T} = I$

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