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Determinant Solver

Determinant Solver

Type values — result updates instantly. Supports 2×2 to 6×6.

Last edited cell: 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).

General formula

Leibniz formula: det(A) = Σσ sgn(σ) Πi ai,σ(i). Practical computation uses elimination/factorization for efficiency.

2×2, 3×3, 4×4

2×2: det([[a,b],[c,d]]) = ad − bc.

3×3:

4×4: Use cofactor expansion or LU; this calculator uses LU under the hood.

Properties

  • → det(AB) = det(A)·det(B)
  • → det(Aᵀ) = det(A)
  • → Swap two rows → sign flips
  • → Scale a row by k → determinant scales by 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

Quick Breakdown

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

How the Determinant Solver Works

This calculator parses your entries and computes the determinant in real time. For numerical stability and performance it uses LU decomposition with partial pivoting. The result equals the product of the diagonal of the U factor times the sign from row swaps.

Frequently Asked Questions

1What is a determinant?
It’s a scalar computed from a square matrix. Geometrically, it scales areas/volumes and encodes orientation. A zero determinant means the matrix is singular (non-invertible).
2General formula
Leibniz formula: det(A) = Σσ sgn(σ) Πi ai,σ(i). Practical computation uses elimination/factorization for efficiency.