Let $\;\{ \vec i,\vec j,\vec k \}\;$ be the standard co-ordinate system of $\;\mathbb R^3\;$ and consider these two matrices:
- $\;A=\begin{pmatrix} 0\;\;0\;-1\\0\;\;\frac{1}{x_2}\;\;0\\1\;\;\;0\;\;\;0\\ \end{pmatrix}\;$
- $\;B\;$ a $\;3\times 3\;$ orthonormal matrix such that $\;B \vec i=\vec y_1\;,\;B \vec j=\vec y_2\;,\;B \vec k=\vec y_3\;$ where $\;\{ y_1,y_2,y_3 \}\;$ is another orthonormal system of $\;\mathbb R^3\;$. Note: The two co-ordinate systems have the same origin.$\;B\;$ represents a rotation matrix. To be more specific, $\;B\;$ is a counter-clockwise rotation by $\;\hat \theta\;$ about the $\;x-axis\;$.
QUESTION: Do $\;A,B^T\;$ commute in the sense that $\;AB^T=B^TA\;$?
My Attempt:
I wrote $\;A\vec i=-\vec k\;,\;A\vec j=1\;,\;A\vec k=\vec i\;$ and thought it is sufficient to show $\;B^TA\vec i=AB^T\vec i\;$,$\;B^TA\vec j=AB^T\vec j\;$,$\;B^TA\vec k=AB^T\vec k\;$. However I got stuck in the very beginning:
$\;B^TA\vec i=B^T(-\vec k)\;$ How should I proceed?
I'm having a really hard time getting my head around this. To be honest, I'm not sure if the matrices commute in first place. And if they do, I don't know how to prove it.
I would appreciate any help. Thanks in advance!