Problems arise when systems are coupled. For example \[\begin{bmatrix} \cdot^2 & 1 \\ 0 & \cdot^2 \end{bmatrix}\odot \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} a \\ b \end{bmatrix}\] expressing \[x^2+y=a\\ y^2=b\\\] we can clearly solve for the inverse, but find \[y=\pm\sqrt{b}\\ x=\pm\sqrt{a\mp\sqrt{b}}\]
In this sense we can’t express the inverse as hoped. But can instead write \[\begin{bmatrix} x \\ y \end{bmatrix} =\begin{bmatrix} \pm\sqrt{\cdot} & 0 \\ 0 & \pm\sqrt{\cdot} \end{bmatrix} \odot \begin{bmatrix} a\mp\sqrt{b} \\ b \end{bmatrix}\] we see the inverse matrix is the same as the uncoupled version above. But we now require an additional matrix transformation to make the equation complete, \[\begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix}\pm\sqrt{\cdot} & 0 \\ 0 & \pm\sqrt{\cdot}\end{bmatrix} \odot\left( \begin{bmatrix}1 & \mp\sqrt{\cdot} \\ 0 & 1 \end{bmatrix} \odot\begin{bmatrix}a \\ b\end{bmatrix}\right)\] Note the parenthesis must be present as there is a non-associativity present