Let $\vec{a}\in R^m$ and $\vec{b}\in R^n$ be two vector functions. Now $\frac{\partial \vec{a}}{\partial \vec{b}}$ is given to us and will be a $m\times n$ matrix.
Here I am assuming that the first row is $\frac{\partial a_1}{\partial b_k}$ where $k=1,\cdots,n$.
Now we want to compute $\frac{\partial \vec{b}}{\partial \vec{a}}$ from the given $m\times n$ matrix. If I take its transpose and then invert each scalar element of the transposed matrix, does the resulting matrix automatically give me my desired $n\times m$ matrix, viz. $\frac{\partial \vec{b}}{\partial \vec{a}}$
No, you need to properly invert the matrix. This will require $n=m$ and the matrix to have full rank.