Defining the determinant行列式ぎょうれつしき by permutations置換ちかん

date2026-07-14document_iddoc_e71393960578c58ac4e3a17955dff07adescription置換と符号を用いて一般の行列式を定義し、多重線型性・交代性・三角行列の公式・積の公式がどこから出るかを証明する講義である。prerequisites正方行列 / 順列と置換の基本 / 線型性 / 行列式の直感type講義content_typelecturestatusactive

mathlinear-algebraundergraduatelecturedeterminant

1Introduction

The key idea is to define the determinant行列式ぎょうれつしき as a signed sum over all ways to choose one entry from each row and one entry from each column. A permutation置換ちかん records such a choice, and its sign records whether the choice preserves or reverses orientation.

2Terms and definitions

A permutation置換ちかん of ${1, \dots, n}$ is a bijection全単射ぜんたんしゃ from the set to itself. The set of all such permutations is denoted by $S_{n}$ .

An inversion反転はんてん of a permutation $σ$ is a pair $i < j$ with $σ (i) > σ (j)$ . The sign is

sgn (σ) = (- 1)^{numberofinversions} .

For an $n \times n$ matrix $A = (a_{i j})$ , define

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

Each term chooses one entry from each row and one entry from each column.

3Plan

We first take the permutation formula as the definition. Then we prove the main properties: normalization, multilinearity, sign change under row or column swaps, vanishing when two rows or columns coincide, uniqueness from these principles, triangular formulas, and multiplicativity.

4Intuitive explanation

Each determinant term chooses one entry from each row and each column. The permutation makes the no-repetition rule explicit. The sign records whether the chosen column order is an even or odd reordering of the standard order.

For a $2 \times 2$ matrix there are only two choices. The identity choice gives $a d$ , and the swapped choice gives $b c$ with negative sign, producing $a d - b c$ .

5Precise explanation

The formula recovers the $2 \times 2$ determinant because the two permutations of ${1, 2}$ give $a d$ and $b c$ with opposite signs.

The identity matrix has determinant $1$ , since only the identity permutation contributes a nonzero term.

The determinant is multilinear in rows and columns because each term contains exactly one factor from each row and each column. Swapping two rows or columns changes the permutation sign, so the determinant changes sign. If two rows or columns are equal, the swap leaves the matrix unchanged but also changes the sign of the determinant, forcing the determinant to be zero over the real or complex numbers.

The permutation formula is also the unique function that is multilinear, alternating, and normalized by $\det I = 1$ . Expanding each row in the standard basis leaves only terms indexed by permutations, and the alternating property supplies the sign.

For triangular matrices, only the identity permutation can contribute a nonzero term, so the determinant is the product of the diagonal entries. The multiplicative formula

\det (A B) = \det A \det B

follows by applying the uniqueness characterization to the function $F (A) = \det (A B)$ .

6Worked example

For

A = (\begin{matrix} 1 & 2 & 0 \\ 0 & 3 & 4 \\ 5 & 0 & 6 \end{matrix}),

the six permutation terms give

\begin{aligned} \det A &= a_{11}a_{22}a_{33} +a_{12}a_{23}a_{31} +a_{13}a_{21}a_{32}\\ &\quad -a_{13}a_{22}a_{31} -a_{12}a_{21}a_{33} -a_{11}a_{23}a_{32}\\ &=58. \end{aligned}\begin{aligned} \det A &= a_{11}a_{22}a_{33} +a_{12}a_{23}a_{31} +a_{13}a_{21}a_{32}\\ &\quad -a_{13}a_{22}a_{31} -a_{12}a_{21}a_{33} -a_{11}a_{23}a_{32}\\ &=58. \end{aligned}

The point is not to memorize six terms mechanically; each term corresponds to one permutation choice.

7Other viewpoints

Algebraically, the permutation formula gives a complete componentwise definition. Geometrically, the sign records oriented volume. Computationally, the formula is usually not efficient for large matrices because it has $n!$ terms; row reduction and cofactor expansion are usually better for calculation.

data/lecture/math/linear-algebra/determinant-computation-rules.lecture.n.md data/lecture/math/linear-algebra/cofactor-expansion-and-invertibility.lecture.n.md

8How to recognize when to use this

Use the permutation definition when proving determinant properties or checking where the signs come from. For numerical computation, use cofactor expansion or row operations unless the matrix is very small.

9Scope

Here the entries are real or complex numbers and the matrix is square. Rectangular matrices do not have this determinant because the permutation choice requires the same number of rows and columns. More general number systems are not needed at this point in the linear algebra sequence.

10Final forms

[PARSE ERROR: Undefined("Command(\"boxed\")")] \det A = \sum_{σ \in S_{n}} sgn (σ) a_{1 σ (1)} a_{2 σ (2)} \dots a_{n σ (n)}

[PARSE ERROR: Undefined("Command(\"boxed\")")] \det I = 1, \det (A B) = \det A \det B

[PARSE ERROR: Undefined("Command(\"boxed\")")] multilinearity a l t e r n a t i o n + n o r m a l i z a t i o n d e t e r m i n e t h e d e t e r m i n a n t [PARSE ERROR: Undefined("RBrace")]

11In one sentence

A permutation records one entry chosen from each row and column.
Its sign records orientation.
The determinant rules follow from this signed sum.

12Exercise links

data/exercise/math/linear-algebra/determinants-and-invertibility.exercise.n.md