How to prove the derivative of $a^{T} A^{-1} b$ with respect to $A$

Question

So I can find from the matrix cookbook here:

$\frac{\partial a^TX^{-1}b}{\partial X} = -X^{-T}ab^TX^{-T}$

To prove it, I have tried expanding:

$a^TX^{-1}b = \sum\limits_{i,j}^{n,n}a_i(X^{-1})_{ij}b_j$

I can also find from cookbook where:

$\frac{\partial (X^{-1})_{ij}}{\partial X_{ij}} = -(X^{-1})_{ij}(X^{-1})_{ij}$

However, what I cannot figure out is that where did that transpose come from?

Any idea and help are appreciated!