how to change polar coordinate into cartesian coordinate using transformation matrix I would like to change $(3,4,12)$ in $xyz$ coordinate to spherical coordinate using the following relation

It is from the this link. I do not understand the significance of this matrix (if not for coordinate transformation) or how it is derived. Also please check my previous question building transformation matrix from spherical to cartesian coordinate system. Please I need your insight on building my concept.
Thank you.
EDIT::
I understand that $ \left [ A_x \sin \theta\cos \phi \hspace{5 mm} A_y \sin \theta\sin\phi \hspace{5 mm} A_z\cos\theta\right ]$ gives $A_r$ but how is other coordinates $ (A_\theta, A_\phi)$ equal to their respective respective rows from Matrix multiplication?

I have checked the formula on the link to transform from cartesian to spherical co-ords and it is correct. While it is correct that this is a nonlinear transformation for a vector field, the formula represent the correct linear transformation of a vector at any particular point in that field. Hope that helps since you helped me to fine that link.
