I was looking for theorems that might be helpful in order for some proofs that I have and I came across the following one:


$$\frac{d}{dt} [\det A(t)]=\det A(t) \cdot \operatorname\*{tr}[A^{-1}(t)\cdot \frac{d}{dt} A(t)]$$


where $A(t)$ is a matrix with a variable $t$.


The problem is that I have neither a reliable source for this theorem nor am I able to prove it.


Did anyone come across the aforementioned equation or is able to prove it?



Proof for the derivative of the determinant of a matrix
This is just Jacobi's formula in the case of $A$ invertible.


Most books with any matrix theory in it should have a proof. Even wikipedia has [one](https://en.wikipedia.org/wiki/Jacobi%27s_formula#Derivation).


