If A= […] is an orthogonal matrix, then |abc| is equal to

If A is an orthogonal matrix, then

$AA^T =I=A^TA$

$⇒\begin{bmatrix}0&2b&c\\a&b&-c\\a&-b&c\end{bmatrix}\begin{bmatrix}0&ab&a\\2b&b&-b\\c&--c&c\end{bmatrix}=\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}$

$⇒\begin{bmatrix}4b^2+ c^2&2b^2-c^2&-2b^2 + c^2\\2b^2-c^2&a^2+b^2+c^2&a^2-b^2-c^2\\-2b^2 + c^2&a^2-b^2-c^2&a^2+b^2+c^2\end{bmatrix}=\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}$

$⇒4b^2+c^2=1,2b^2-c^2=0$ and $a^2-b^2-c^2=0$

$⇒a^2=\frac{1}{2},b^2=\frac{1}{6}$ and $c^2=\frac{1}{3}⇒|abc|=\frac{1}{6}$