If the following transformation matrix of the components of a vector $\vec{A}$ from cartesian to polar coordinates is true:
$$ \begin{bmatrix}A_x\\A_y\end{bmatrix} = \begin{bmatrix}\cos \theta &-r\sin \theta \\\sin \theta &r\cos \theta \end{bmatrix} \begin{bmatrix}A_r\\A_{\theta} \end{bmatrix} $$
(I found it here: https://math.stackexchange.com/q/4325357), then I have a trouble in understanding the expression for the length of $|\vec{A}|$ in polar coordinates. If I observe that
$$ P = \begin{bmatrix}\cos \theta &-r\sin \theta \\\sin \theta &r\cos \theta \end{bmatrix} =\begin{bmatrix}\cos \theta &-\sin \theta \\\sin \theta &\cos \theta \end{bmatrix} \begin{bmatrix} \ 1 & 0\\\ 0 & r\end{bmatrix} == R \cdot g $$
then I easily calculate $$ \begin{bmatrix} \ A_x & A_y \end{bmatrix} \cdot \begin{bmatrix} \ A_x \\\ A_y \end{bmatrix} = \begin{bmatrix} \ A_r & A_\theta \end{bmatrix} g^{T} R^{T} R g \begin{bmatrix} \ A_r \\\ A_\theta \end{bmatrix} = A_r^2 + r^2 A_\theta^2 = |\vec{A}|$$
What on earth does this formula $|\vec{A}| = A_r^2 + r^2 A_\theta^2 $ mean? It resembles the infinitesimal line element in polar coordinates, but I don't see the meaning. Please clarify!