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行列式ソルバー

行列式ソルバー

2×2、3×3、4×4、5×5、6×6 行列の行列式を即時計算。リアルタイム結果、すっきりUI、性質、公式、FAQ。モバイル対応。

最後に編集したセル: a[1,1]

What is a determinant?

A determinant is a special scalar value calculated from a square matrix. It tells you if the matrix is invertible, how it scales areas/volumes, and encodes orientation. A zero determinant means the matrix is singular (non-invertible).

一般式

ライプニッツの公式: det(A) = Σσ sgn(σ) Πi ai,σ(i). 実用的には効率のため消去法/分解を用います。

2×2、3×3、4×4

2×2: det([[a,b],[c,d]]) = ad − bc。

3×3:

4×4: 余因子展開やLU分解を使用します。この計算機は内部でLU分解を使います。

性質

  • → det(AB) = det(A)·det(B)
  • → det(Aᵀ) = det(A)
  • → 2行を入れ替える → 符号が反転
  • → ある行をk倍 → 行列式もk倍

Understanding Determinants & Matrix Determinant Calculations

A complete guide to calculating matrix determinants using our Determinant Solver

What is a Determinant?

A determinant is a special scalar value that can be calculated from any square matrix (a matrix with equal rows and columns). Written as det(A) or |A|, it encapsulates fundamental properties of the matrix in a single number.

The determinant tells you whether a system of linear equations has a unique solution, whether a matrix can be inverted, and how transformations affect geometric objects like area and volume.

Geometric Interpretation

2D: Area Scaling

The absolute value of a 2×2 determinant equals the area of the parallelogram formed by the row vectors.

3D: Volume Scaling

For 3×3 matrices, the determinant represents the volume of the parallelepiped spanned by the row vectors.

Sign = Orientation

Positive determinant preserves orientation; negative means reflection. Zero means the space is "flattened."

Why Determinants Matter

📊

Solving Linear Systems

Cramer's Rule uses determinants to solve systems of equations. If det ≠ 0, a unique solution exists.

🔄

Matrix Invertibility

A matrix has an inverse if and only if its determinant is non-zero. This is fundamental in linear algebra.

📐

Eigenvalue Problems

Finding eigenvalues requires solving det(A − λI) = 0, essential in physics and engineering.

🌐

Computer Graphics

Transformations in 3D graphics use matrices. The determinant reveals if shapes are flipped or scaled.

Zero vs Non-Zero Determinant

det(A) ≠ 0

  • ✓ Matrix is invertible
  • ✓ Rows/columns are linearly independent
  • ✓ System Ax = b has unique solution
  • ✓ Full rank matrix

det(A) = 0

  • ✗ Matrix is singular (non-invertible)
  • ✗ Rows/columns are dependent
  • ✗ System has no or infinite solutions
  • ✗ Rank deficient matrix

クイック解説

See It In Action

Watch how determinants are calculated step-by-step with interactive examples

2×2 Matrix
3
8
4
6
=
-14
3×6 − 8×4 = -14
3×3 Sarrus Rule
1
2
3
4
5
6
7
8
9
=
0
Singular matrix (det = 0)
Identity Matrix
1
0
0
0
1
0
0
0
1
=
1
Identity always = 1

Interactive Step-by-Step: 2×2 Determinant

Step 1
Start with matrix
a b c d
Step 2
Multiply diagonals
a × d = ad
b × c = bc
Step 3
Subtract products
ad − bc
Result
Determinant
det(A)
= ad − bc

Instant Results

Calculations update in real-time as you type

Flexible Sizes

Support for 2×2 up to 6×6 matrices

Step Breakdown

See the calculation process explained

Easy Export

Copy results or matrices with one click

行列式ソルバーの仕組み

入力値を解析し、リアルタイムで行列式を計算します。数値安定性と速度のため、部分ピボット付きLU分解を使用します。結果はUの対角成分の積に、行交換による符号を掛けたものです。

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よくある質問

1行列式とは？
正方行列から計算されるスカラー値です。幾何学的には面積/体積の倍率や向きを表します。行列式が0なら行列は特異（逆行列を持たない）です。
2一般式
ライプニッツの公式: det(A) = Σσ sgn(σ) Πi a_{i,σ(i)}。実用的には効率のため消去法/分解を用います。
⚡

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