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

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

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

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?

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

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

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?

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?

Post Closed as "Not suitable for this site" by Suvrit, Boris Bukh, Alex Degtyarev, Chris Godsil, Will Sawin
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Proof for the derivative of the determinant of a matrix

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

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

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?