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。
- 整数・小数・科学表記（例: 1e-3）に対応。
- 2×2〜6×6に対応。より大きいサイズも可能ですがモバイル表示のため非表示です。
- 単位行列/ランダムボタンで素早くテストできます。
- 計算値を変えずに表示桁数を調整できます。
行列式ソルバー
値を入力すると結果が即時更新されます。2×2〜6×6に対応。
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
A complete guide to one of the most important concepts in linear algebra
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
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。