---
tag: 0ZN2J42SK7
kind: statement
role: proposition
name: Determinant of a 2 by 2 matrix
msc: [15A15]
uses: [0C3CFHDN6R]
sources:
  - url: https://en.wikipedia.org/wiki/Determinant
    locator: Two by two matrices
---

## Statement

Let $K$ be a commutative ring and let $a, b, c, d \in K$.
Then
$$\det \begin{pmatrix} a & b \\ c & d \end{pmatrix} = ad - bc,$$
where $\det$ is the [[determinant|00KDZRW4C0]].

## Remarks

This is the [[Leibniz formula|016PE4VZ8Q]] for $n = 2$, with the identity and one transposition.
Over $\R$, $\abs{ad - bc}$ is the area of the parallelogram spanned by the columns $(a, c)$ and $(b, d)$.
For example, $\det \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} = 4 - 6 = -2$.

## Explanations

### L1 human

The determinant of a $2 \times 2$ matrix is the product of the main diagonal minus the product of the other diagonal.
It is zero exactly when one column is a multiple of the other, over a field.
Then the matrix squeezes the plane onto a line or a point.