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?

The transformation from Cartesian to polar coordinates is not a linear function, so it cannot be achieved by means of a matrix multiplication.

This is not the Matrix you're looking for. For a simple co-ordinate switch you can just use the relations:

$$\begin{align*}x &= \rho\sin\theta\cos\phi\\ y &= \rho\sin\theta\sin\phi \\ z &= \rho\cos\theta\end{align*}$$

And the inverse operations:

$$\begin{align*}\rho &= \sqrt{x^2 + y^2 + z^2}\\ \phi &= \arctan\dfrac yx\\ \theta &= \arctan\left(\frac{\sqrt{x^2 + y^2}}z\right)\end{align*}$$

However the matrix you've found is for mapping a vector between the co-ordinate systems. For example (using a textbook, Engineering Electromagnetics by Demarest. Example 2-6, p34)

Need to do an integration of $\int( r^3\cos\phi\sin\theta\cdot Ar) d\theta d\phi$

Where $Ar$ is a unit vector in the radial direction. The integral is over phi and theta but also dependent on phi and theta, therefore it's much easier to do this by switching back to cartesian coordinates by the relation:

$$Ar = \sin\theta\cos\phi\cdot Ax + \sin\theta\sin\phi\cdot Ay + \cos\theta\cdot Az$$

Once we substitute that straight in for Ar the integral looks longer but we've removed the dependence inside the integrand, so we can do the integration in a straight forward way.