The values of x,y,z if the matrix $A = [\begin{matrix} 0 & 2 y & z \\ x & y & - z \\ x & - y & z \end{matrix}]$ satisfy the equation $A^{T} A = I$ are

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1, 2, 3

$\pm \frac{1}{\sqrt{2}}, \pm \frac{1}{\sqrt{6}}, \pm \frac{1}{\sqrt{3}}$

$\pm \frac{1}{\sqrt{3}}, \pm \frac{1}{\sqrt{6}}, \pm \frac{1}{\sqrt{2}}$

$\pm \frac{1}{\sqrt{6}}, \pm \frac{1}{\sqrt{3}}, \pm \frac{1}{\sqrt{2}}$

answer is A.

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Detailed Solution

$\begin{array}{rcl} A A^{T} & = & I \\ [\begin{matrix} 0 & 2 y & z \\ x & y & - z \\ x & - y & z \end{matrix}] [\begin{matrix} 0 & x & x \\ 2 y & y & - y \\ z & - z & z \end{matrix}] & = & [\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}] \\ [\begin{matrix} 4 y^{2} + z^{2} & 2 y^{2} - z^{2} & - 2 y^{2} + z^{2} \\ 2 y^{2} - z^{2} & x^{2} + y^{2} + z^{2} & x^{2} - y^{2} - z^{2} \\ - 2 y^{2} + z^{2} & x^{2} - y^{2} - z^{2} & x^{2} + y^{2} + z^{2} \end{matrix}] & = & [\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}] \\ clearly x & = & \pm \frac{1}{\sqrt{2}}, y = \pm \frac{1}{\sqrt{6}}, z = \pm \frac{1}{\sqrt{3}} \end{array}$