---
tag: 357FEAZ0Q4
kind: statement
role: theorem
name: Jacobi's formula
msc: [15A15]
uses: [00KDZRW4C0, 07W9H8QF38, 0XC1RDBY7A, 0K57YG3BB8, 0XPKR18APA, 0C08BQ3M05]
about: [00KDZRW4C0]
sources:
  - url: https://en.wikipedia.org/wiki/Jacobi%27s_formula
same_as:
  wikidata: Q3748375
  wikipedia: Jacobi's formula
---

## Statement

Let $\mathbb{K}$ be $\R$ or $\C$, let $I \subseteq \R$ be an open interval, and let $A \colon I \to \mathbb{K}^{n \times n}$ be a [[differentiable|0K57YG3BB8]] map.
Then $t \mapsto \det A(t)$ is differentiable, and for all $t \in I$
$$\frac{d}{dt} \det A(t) = \tr\big(\operatorname{adj}(A(t))\, A'(t)\big),$$
where $\det$ is the [[determinant|00KDZRW4C0]], $\operatorname{adj}$ is the [[adjugate|07W9H8QF38]] and $\tr$ is the [[trace|0XC1RDBY7A]].
If $A(t)$ is invertible, this equals $\det A(t) \cdot \tr\big(A(t)^{-1} A'(t)\big)$.

## Remarks

The determinant is a polynomial in the entries $a_{ij}$.
By the [[Laplace expansion|0C08BQ3M05]] along row $i$, its partial derivative with respect to $a_{ij}$ is the [[cofactor|0XPKR18APA]] $C_{ij}$.
The chain rule gives $\frac{d}{dt}\det A = \sum_{i,j} C_{ij}\, a_{ij}'$.
This sum is $\tr(\operatorname{adj}(A) A')$, because $\operatorname{adj}(A)_{ji} = C_{ij}$.
The second form follows from $\operatorname{adj}(A) = \det(A) A^{-1}$.

A special case is $\frac{d}{dt}\det(I + tX)\big|_{t=0} = \tr X$: the derivative of the determinant at the identity is the trace.
For $A(t) = e^{tX}$, the formula gives $\frac{d}{dt}\det e^{tX} = \tr(X) \det e^{tX}$, so $\det e^{tX} = e^{t \tr X}$.

## Explanations

### L1 human

Jacobi's formula tells how fast the determinant of a matrix changes when the matrix changes.
Near the identity matrix, the rate of change of the determinant is the trace of the change.
For example, $\det\begin{pmatrix} 1 + t & 0 \\ 0 & 1 + 2t \end{pmatrix} = 1 + 3t + 2t^2$ has derivative $3$ at $t = 0$.
This is the trace of $\begin{pmatrix} 1 & 0 \\ 0 & 2 \end{pmatrix}$.

### L2 human

The formula says that the differential of $\det$ at $A$ is the linear map $X \mapsto \tr(\operatorname{adj}(A) X)$.
It gives Liouville's formula for the Wronskian of a linear system of differential equations.
It also gives the [[determinant of a matrix exponential|0A1QQN333A]], $\det e^A = e^{\tr A}$.
In Lie theory it says that the derivative of $\det \colon \mathrm{GL}_n(\mathbb{K}) \to \mathbb{K}^\times$ at the identity is the trace.