What is the angle between a $A_{3x3}$ with $Rank(A)=2$ and $A^T$

Question

I need to solve this problem:

For Matrix $A_{3x3}$ with $Rank(A)=2$. If Matrix A is Transposed and its elements are the same as elements of Matrix B. What is the angle of rotation from A to B?

Answer 1

This is what I have so far:

I started by defining $A=\begin{pmatrix}a&b&c\\d&e&f\\0&0&0\end{pmatrix}$ and $B=A^T=\begin{pmatrix}a&d&0\\b&e&0\\c&f&0\end{pmatrix}$.

$<A,B>=Trace(B^TA)=Trace(AA)=a^2+2bd+e^2$

$\Vert A \Vert=\sqrt{<A,A>}=\sqrt{Trace(A^TA)}=\sqrt{Trace(BA)}=\sqrt{a^2+b^2+c^2+d^2+e^2+f^2}$

$\Vert B \Vert=\sqrt{<B,B>}=\sqrt{Trace(B^TB)}=\sqrt{Trace(AB)}=\sqrt{a^2+b^2+c^2+d^2+e^2+f^2}$

$\theta={cos}^{-1}(\frac{<A,B>}{\Vert A \Vert \Vert B \Vert})={cos}^{-1}(\frac{a^2+2bd+e^2}{a^2+b^2+c^2+d^2+e^2+f^2})$

What's the significance of this result? Or is there any other way of approaching this? Because I'm lost.