Derivative of matrix determinant wrt to matrix element

Question

I want to take the derivative of the function:

$$ f(A) = 2 + \text{log}(|\text{det}(A)|) $$

with respect to the matrix element $A_{ij}$, where A is orthogonal, so far:

\begin{align*} \frac{\partial}{\partial A_{ij}}f(A) &= \frac{\partial}{\partial A_{ij}} \text{log}(|\text{det}(A)|)\\ & = \frac{\frac{\partial}{\partial A_{ij}}|\text{det(A)}|}{|\text{det(A)}|} \\ &=\frac{\partial}{\partial A_{ij}}|\text{det(A)}| \end{align*}

I'm not really sure how to proceed, my first instinct was to note that $det(A) = I$ for an orthogonal matrix, but i guess here I am trying to find how the determinant changes wrt a matrix element.

Answer 1

You can use Jacobi's identity

\begin{eqnarray} \frac{\partial}{\partial A_{ij}}f(A) &=& \frac{\partial }{\partial A_{ij}} \left( 2 + \log |\det(A)|\right) \\ &=& \frac{1}{|\det(A)|}\frac{\partial}{\partial A_{ij}}\det (A) = C_{ij}(A) \end{eqnarray}

where $C(A)$ is the cofactor matrix of A