16.5 Permutation Definition of Determinant

Properties: 16.6 Rule of Sarrus,
Sufficiencies: 16.2 Symmetric Groups
Questions: N/A

Permutation Definition of Determinant

Let $A \in R^{n \times n}$ . Then there is a unique determinant function

det : R^{n \times n} \to R

that maps a square matrix to a single scalar value given by

det (A) = \sum_{π \in S_{n}} sgn (π) \cdot a_{1 π (1)} \cdot a_{2 π (2)} \dots a_{n π (n)}

Remark. A function $det : R^{n \times n} \to R$ is a determinant function iff it meets the following conditions:

$⧫$

Let's look at how this definition applies to a $2 \times 2$ matrix.

Determinant of

A \in R^{2 \times 2}

Let $A \in R^{2 \times 2}$ with

A = [\begin{matrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{matrix}]

Let's find the determinant of $A$ using our definition.

\begin{aligned} det (A) & = \sum_{n \in S_{2}} sgn (π) \cdot a_{1 π (1)} \cdot a_{2 π (2)} \end{aligned}

$S_{2}$ is our 16.2 Symmetric Group acting on a set with 2 elements.

S_{2} = {π : {1, 2} \to {1, 2} : π is bijective}

Using Cauchy's two-line notation, we write possible permutations as

\begin{aligned} π_{1} := (\begin{array}{c} 1 & 2 \\ 1 & 2 \end{array}), & π_{2} := \overset{inputs}{\underset{outputs}{(\begin{array}{c} 1 & 2 \\ 2 & 1 \end{array})}} \end{aligned}

To find the sign of our permutations, consider:
![[Sign of Permutations.pdf]]
For any $2 \times 2$ matrix, we see that

\begin{aligned} ⟹ det (A) & = \sum_{n \in S_{2}} sgn (π_{i}) \cdot a_{1 π_{i} (1)} \cdot a_{2 π_{i} (2)} \\ = \sum_{i = 1}^{2} sgn (π_{i}) \cdot a_{1 π_{1} (1)} \cdot a_{2 π_{2} (2)} \\ = \underset{j = 1}{\underset{⏟}{sgn (π_{1}) \cdot a_{1 π_{1} (1)} \cdot a_{2 π_{1} (2)}}} + \underset{j = 2}{\underset{⏟}{sgn (π_{2}) \cdot a_{1 π_{2} (1)} \cdot a_{2 π_{2} (2)}}} \\ = + 1 \cdot a_{11} \cdot a_{22} + (- 1) \cdot a_{12} \cdot a_{21} \\ = a_{11} \cdot a_{22} - a_{12} \cdot a_{21} \end{aligned}