---
tag: 0Z4BWARC38
kind: statement
role: proposition
name: Determinant of the identity matrix
msc: [15A15]
uses: [0C3CFHDN6R, 06DW9BAWB1]
sources:
  - url: https://en.wikipedia.org/wiki/Determinant
    locator: Basic rules
---

## Statement

Let $K$ be a commutative ring and let $n \ge 1$.
Then $\det I_n = 1$, where $I_n$ is the [[identity matrix|06DW9BAWB1]] and $\det$ is the [[determinant|00KDZRW4C0]].

## Remarks

This is property 3 of the definition of the determinant.
It fixes the scale: the unit cube has volume $1$.
More generally, the determinant of a diagonal matrix is the product of its diagonal entries.

## Explanations

### L1 human

The identity matrix leaves every vector where it is.
It does not change any volume.
So its determinant is $1$.