Contravariant Vector Component Transformation from Polar to Cartesian

Question

I am new to tensors and I just learned that the contravarient components of a vector transforms in the following way (using Einstein summation convention) $$A^{'i}=\frac {\partial x^{'i}}{\partial x^j}A^j$$ I want to transform the components of a vector from polar $(r,\theta)$ coordinate to cartesian $(x,y)$ coordinate. So, I use $x^{'1}=x, x^{'2}=y, x^1=r$ and $x^2=\theta$. Also, $x=r\cos\theta$ and $y=r\sin\theta$. After performing the partial derivatives, I obtain the transformation matrix as $$ \begin{pmatrix} \cos\theta & -r\sin\theta \\ \sin\theta & r\cos\theta \\ \end{pmatrix} $$ But, I know from my basic vector studies that the component transformation matrix from polar to cartesian is $$ \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \\ \end{pmatrix} $$ Can anyone please explain why this discrepancy arises? Have I misunderstood something?