Practicing Success
If $A=\begin{bmatrix}0&2b&c\\a&b&-c\\a&-b&c\end{bmatrix}$ is an orthogonal matrix, then $|abc|$ is equal to |
$\frac{1}{2}$ $\frac{1}{3}$ $\frac{1}{6}$ 1 |
$\frac{1}{6}$ |
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 $c^2=\frac{1}{3}⇒|abc|=\frac{1}{6}$ |