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Kalkulator Determinan

Kalkulator Determinan

Ketik nilai — hasil langsung diperbarui. Mendukung 2×2 hingga 6×6.

Sel terakhir diedit: 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).

Rumus umum

Rumus Leibniz: det(A) = Σσ sgn(σ) Πi ai,σ(i). Perhitungan praktis menggunakan eliminasi/faktorisasi untuk efisiensi.

2×2, 3×3, 4×4

2×2: det([[a,b],[c,d]]) = ad − bc.

3×3:

4×4: Gunakan kofaktor atau LU; kalkulator ini memakai LU di balik layar.

Properti

  • → det(AB) = det(A)·det(B)
  • → det(Aᵀ) = det(A)
  • → Tukar dua baris → tanda berubah
  • → Skalakan satu baris dengan k → determinan ikut dikalikan k

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

Ringkasan cepat

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

Cara kerja kalkulator determinan

Kalkulator ini membaca input Anda dan menghitung determinan secara real-time. Untuk stabilitas numerik dan kinerja, digunakan dekomposisi LU dengan pivot parsial. Hasilnya adalah produk diagonal U dikalikan tanda dari pertukaran baris.

Pertanyaan yang Sering Diajukan

1Apa itu determinan?
Ini adalah skalar yang dihitung dari matriks persegi. Secara geometris, ia menskalakan luas/volume dan mengodekan orientasi. Determinan nol berarti matriks singular (tidak dapat diinvers).
2Rumus umum
Rumus Leibniz: det(A) = Σσ sgn(σ) Πi a_{i,σ(i)}. Dalam praktiknya digunakan eliminasi/faktorisasi untuk efisiensi.