---
tag: 0A1QQN333A
kind: statement
role: theorem
name: Determinant of a matrix exponential
msc: [15A16, 15A15]
uses: [0K665H5V1S, 0C3CFHDN6R, 07PTPSZPME]
sources:
  - url: https://en.wikipedia.org/wiki/Matrix_exponential
    locator: The determinant of the matrix exponential
---

## Statement

Let $\mathbb{K}$ be $\R$ or $\C$, and let $A$ be an $n \times n$ matrix over $\mathbb{K}$.
Then
$$\det(e^A) = e^{\tr A},$$
where $e^A$ is the [[matrix exponential|020M04FN1H]], $\det$ is the [[determinant|00KDZRW4C0]] and $\tr$ is the [[trace|0XC1RDBY7A]].

## Remarks

If $A$ is triangular over $\C$ with diagonal entries $\lambda_i$, then $e^A$ is triangular with diagonal entries $e^{\lambda_i}$, and both sides equal $e^{\lambda_1 + \cdots + \lambda_n}$.
The general case follows by [[triangularisation|0363JMQ42Y]] over $\C$.
It shows again that $e^A$ is invertible, because $e^{\tr A} \ne 0$.
If $\tr A = 0$, then $e^A$ lies in the special linear group.

## Explanations

### L1 human

The determinant of the exponential of a matrix is the exponential of its trace.
This links the product of the eigenvalues of $e^A$ with the sum of the eigenvalues of $A$.
It is the matrix form of the rule $e^{a} e^{b} = e^{a + b}$.