Determinant行列式ぎょうれつしき

1Introduction

The determinant行列式ぎょうれつしき is a number attached to a square matrix正方行列せいほうぎょうれつ. It measures the oriented scale factor of area or volume and detects whether a square linear map線型写像せんけいしゃぞう is invertible可逆かぎゃく.

2Terms and definitions

For a $2\times 2$2\times2 matrix,

the determinant行列式ぎょうれつしき is

For an $n\times n$n\times n matrix, the determinant行列式ぎょうれつしき is characterized by multilinearity in rows or columns, alternation when two rows or columns coincide, and $\det I=1$\det I=1.

3Plan

In dimension $2$2, we introduce the determinant行列式ぎょうれつしき from the area of a parallelogram, and then connect it to systems of linear equations連立一次方程式れんりついちじほうていしき and inverse matrices逆行列ぎゃくぎょうれつ. In higher dimensions, we characterize it by the defining principles of multilinearity多重線型性たじゅうせんけいせい, alternation, and normalization.

4Intuitive explanation

Putting two column vectors列れつベクトル side by side determines a parallelogram. The absolute value of the determinant行列式ぎょうれつしき is its area. More generally, in $R^2$\mathbb R^2, $\left|\det A\right|$|\det A| is the area scale factor under the linear map線型写像せんけいしゃぞう $x\mapsto Ax$x\mapsto Ax. In $R^3$\mathbb R^3, it is the volume scale factor. The sign records orientation: a negative determinant行列式ぎょうれつしき reverses orientation, while a positive determinant行列式ぎょうれつしき preserves it.

Therefore, $\det A=0$\det A=0 means that area or volume collapses to zero. In dimension $2$2, the two columns列れつ collapse onto a single line. Some direction has been lost, so the map cannot be inverted.

5Worked examples

For

we have

This can be read as preserving area scale by a factor of $1$1 while changing the shape of the unit square.

For

we have

The columns are proportional, so the plane collapses to a line and the transformation cannot be inverted.

6Precise explanation

6.11. The $2$2 by $2$2 formula

For

we have

6.22. Why zero is special

If $\det A\ne 0$\det A\ne0, then $A^{-1}$A^{-1} exists and

Thus when $\det A=0$\det A=0, the denominator is zero and the inverse matrix cannot be constructed.

6.33. Concrete examples

For

This can be interpreted as preserving area while changing shape. On the other hand,

has

Its columns are proportional, so the plane is collapsed onto a line.

7High-dimensional definition principle

For a general $n\times n$n\times n matrix, the determinant行列式ぎょうれつしき is not merely a formal extension of the $2\times 2$2\times2 formula $ad-bc$ad-bc. It is the unique function of the columns that satisfies three principles: multilinearity in each column, alternation when two columns are equal or exchanged, and normalization by $\det I=1$\det I=1.

Equivalently,

Here $S_n$S_n is the set of all permutations of $\{1,\dots ,n\}$\{1,\ldots,n\}, and $\mathrm{sgn}\left(\sigma \right)$\operatorname{sgn}(\sigma) is the sign of the permutation. This formula encodes both multilinearity and alternation. The details are developed in the permutation-definition lecture.

8Another viewpoint

Geometrically, the determinant行列式ぎょうれつしき is the area or volume scale factor. Algebraically, it is the existence criterion for an inverse matrix逆行列ぎゃくぎょうれつ. From the viewpoint of systems of linear equations, it tests whether a solution is determined uniquely.

9Criteria

• Use the determinant行列式ぎょうれつしき when invertibility, a unique solution of a square system, or area and volume scaling is at issue.
• The determinant行列式ぎょうれつしき is also useful as a numerical test for whether the column vectors of a square matrix正方行列せいほうぎょうれつ are independent.

10Scope and limitations

In dimensions $2$2 and $3$3, formulas can be written directly. In higher dimensions, permutations or cofactor expansion provide definitions and computations. The essential meaning remains the same: determinant行列式ぎょうれつしき measures volume scaling and detects invertibility.

11Theorem preview: determinant行列式ぎょうれつしき and invertibility

For an $n\times n$n\times n matrix $A$A,

At this point, read this as a theorem to be proved in the following lectures. Row reduction proves it after determinant changes under elementary operations have been established. Cofactors prove it by constructing

when $\det A\ne 0$\det A\ne0.

12Common misunderstandings

• Determinants are defined for square matrices, not arbitrary rectangular matrices.
• The determinant行列式ぎょうれつしき is not just a formula; it records invertibility and volume scaling.
• A zero determinant行列式ぎょうれつしき is structural information, not a calculation failure.

13Final forms

14In one sentence

• The determinant行列式ぎょうれつしき is a number that expresses how much a transformation collapses or stretches space.
• When it becomes $0$0, the transformation cannot be reversed.

15Exercise links

16Related links