which is the direct and inverse transformation matrix?

Direct transformation: $$ dx_1' = \frac{\partial x_1'}{\partial x_1}dx_1 + \frac{\partial x_1'}{\partial x_2}dx_2 \\ dx_2' = \frac{\partial x_2'}{\partial x_1}dx_1 + \frac{\partial x_2'}{\partial x_2}dx_2 $$ $$ \begin{bmatrix} dx_1' \\ dx_2' \end{bmatrix} = \large \begin{bmatrix} \frac{\partial x_1'}{\partial x_1} & \frac{\partial x_1'}{\partial x_2} \\ \frac{\partial x_2'}{\partial x_1} & \frac{\partial x_2'}{\partial x_2} \end{bmatrix} \normalsize \begin{bmatrix} dx_1 \\ dx_2 \end{bmatrix} $$ Inverse transformation: $$ dx_1 = \frac{\partial x_1}{\partial x_1'}dx_1' + \frac{\partial x_1}{\partial x_2'}dx_2' \\ dx_2 = \frac{\partial x_2}{\partial x_1'}dx_1' + \frac{\partial x_2}{\partial x_2'}dx_2' $$ $$ \begin{bmatrix} dx_1 \\ dx_2 \end{bmatrix} = \large \begin{bmatrix} \frac{\partial x_1}{\partial x_1'} & \frac{\partial x_1}{\partial x_2'} \\ \frac{\partial x_2}{\partial x_1'} & \frac{\partial x_2}{\partial x_2'} \end{bmatrix} \normalsize \begin{bmatrix} dx_1' \\ dx_2' \end{bmatrix} $$ With the direct transformation, calculating the inverse results in: $$ \begin{bmatrix} dx_1 \\ dx_2 \end{bmatrix} = \large \begin{bmatrix} \frac{\partial x_2'}{\partial x_2} & -\frac{\partial x_1'}{\partial x_2} \\ -\frac{\partial x_2'}{\partial x_1} & \frac{\partial x_1'}{\partial x_1} \end{bmatrix} / \normalsize \det \begin{bmatrix} dx_1' \\ dx_2' \end{bmatrix} \\ \mbox{with} \qquad \det = \frac{\partial x_1'}{\partial x_1}\frac{\partial x_2'}{\partial x_2} - \frac{\partial x_2'}{\partial x_1}\frac{\partial x_1'}{\partial x_2} $$ With the inverse transformation, calculating the inverse results in: $$ \begin{bmatrix} dx_1' \\ dx_2' \end{bmatrix} = \large \begin{bmatrix} \frac{\partial x_2}{\partial x_2'} & -\frac{\partial x_1}{\partial x_2'} \\ -\frac{\partial x_2}{\partial x_1'} & \frac{\partial x_1}{\partial x_1'} \end{bmatrix} / \normalsize \det' \begin{bmatrix} dx_1 \\ dx_2 \end{bmatrix} \\ \mbox{with} \qquad \det' = \frac{\partial x_1}{\partial x_1'}\frac{\partial x_2}{\partial x_2'} - \frac{\partial x_2}{\partial x_1'}\frac{\partial x_1}{\partial x_2'} $$ Hope you can take it from here.