---
tag: KZQQBSRJMS
kind: proof
name: Proof that the determinant of the identity matrix is one
msc: [15A15]
proves: 0Z4BWARC38
uses: [0C3CFHDN6R, 06DW9BAWB1]
sources:
  - url: https://en.wikipedia.org/wiki/Determinant
    locator: As an alternating multilinear form
---

## Proof

1. Let $I_n$ be the $n \times n$ [[identity matrix|06DW9BAWB1]] over $K$.
2. By property 3 of the [[definition of the determinant|0C3CFHDN6R]], $\det I_n = 1$.

## Idea

The determinant is fixed by three properties, and the value $1$ on the identity is one of them. This normalization picks one function out of all alternating multilinear functions, which differ from it by a constant factor.