Determinant Calculator
Calculate the determinant of 2x2, 3x3, and 4x4 matrices. Find matrix determinants with step-by-step solutions for linear algebra problems.
Enter Matrix Values
Determinant
-2.000000
Matrix Invertible?
Yes
Absolute Value
2.000000
Step-by-Step Solution
Expanding along first row:
det(A) = a₁₁(a₂₂a₃₃ - a₂₃a₃₂) - a₁₂(a₂₁a₃₃ - a₂₃a₃₁) + a₁₃(a₂₁a₃₂ - a₂₂a₃₁)
= 1(-12) - 2(-15) + 5(-4)
= -12 - -30 + -20
= -2
Interpretation
- • det ≠ 0: Matrix is invertible
- • |det| = 2.0000 = volume of transformed unit cube
- • Negative: orientation is reversed
Properties
- • det(AB) = det(A) × det(B)
- • det(A⁻¹) = 1/det(A) = -0.500000
- • det(kA) = k^n × det(A) where n = matrix size
- • det(Aᵀ) = det(A)
Related Calculators
About This Calculator
The determinant is a scalar value that can be computed from the elements of a square matrix. It provides crucial information about the matrix, including whether it's invertible and how it transforms space. This calculator computes determinants for 2×2, 3×3, and 4×4 matrices with step-by-step solutions.
What is a Determinant? The determinant (det) of a square matrix is a single number that encodes important properties of the matrix. For a 2×2 matrix, det = ad - bc. For larger matrices, we use cofactor expansion along a row or column.
Key Uses of Determinants:
- Determining if a matrix is invertible (det ≠ 0)
- Solving systems of linear equations (Cramer's Rule)
- Calculating area and volume transformations
- Finding eigenvalues of matrices
- Understanding linear transformations
Why Determinants Matter:
- Engineering: Structural analysis, circuit theory
- Computer graphics: Transformations, rotations
- Economics: Input-output models
- Physics: Quantum mechanics, relativity
This calculator handles matrices up to 4×4 with detailed solution steps. For matrix operations, see our Matrix Calculator and System of Equations Solver.
How to Use the Determinant Calculator
- 1Select your matrix size (2×2, 3×3, or 4×4).
- 2Enter the matrix elements in the input grid.
- 3Elements are numbered by row and column (a₁₁ is top-left).
- 4Use decimal numbers or negative values as needed.
- 5The determinant calculates automatically as you type.
- 6Review the step-by-step solution below the result.
- 7Check if the matrix is invertible (det ≠ 0).
- 8Note the geometric interpretation (area/volume scaling).
- 9Use the determinant sign to check orientation.
- 10Apply results to your linear algebra problems.
2×2 Determinant Formula
The simplest determinant calculation.
The Formula
For matrix A = [a b; c d]:
det(A) = ad - bc
Visual Pattern
|a b| |c d| = ad - bc
Multiply diagonals: main diagonal minus anti-diagonal.
Example
|3 7| |2 5| = (3)(5) - (7)(2) = 15 - 14 = 1
Geometric Meaning
The absolute value |det| equals the area of the parallelogram formed by the column vectors.
For A = [3 7; 2 5]:
- Column 1: vector (3, 2)
- Column 2: vector (7, 5)
- Parallelogram area = |1| = 1
Special Cases
- det = 0: Columns are parallel (collinear vectors)
- det < 0: Orientation is reversed (reflection)
- det = 1: Area-preserving transformation
3×3 Determinant Methods
Two common approaches for 3×3 matrices.
Cofactor Expansion (First Row)
|a₁ b₁ c₁| |a₂ b₂ c₂| = a₁|b₂ c₂| - b₁|a₂ c₂| + c₁|a₂ b₂| |a₃ b₃ c₃| |b₃ c₃| |a₃ c₃| |a₃ b₃|
Each 2×2 determinant is a "minor."
Sarrus' Rule (Shortcut)
Copy the first two columns to the right:
|a b c|a b |d e f|d e |g h i|g h
Add products of down-diagonals. Subtract products of up-diagonals.
det = aei + bfg + cdh - ceg - afh - bdi
Example
|1 2 3| |4 5 6| |7 8 9|
= 1(45-48) - 2(36-42) + 3(32-35) = 1(-3) - 2(-6) + 3(-3) = -3 + 12 - 9 = 0
This matrix is singular (not invertible).
4×4 and Larger Determinants
Cofactor expansion extends to any size.
Cofactor Expansion Method
Choose a row or column (preferably with zeros). For each element:
- Find the minor (determinant of submatrix)
- Multiply by the element
- Apply checkerboard sign pattern (+, -, +, -, ...)
Sign Pattern
|+ - + -| |- + - +| |+ - + -| |- + - +|
Example Row Expansion
For 4×4 matrix, expand along row 1:
det(A) = a₁₁C₁₁ - a₁₂C₁₂ + a₁₃C₁₃ - a₁₄C₁₄
Where Cᵢⱼ is the 3×3 minor determinant.
Computational Complexity
- 2×2: 2 multiplications
- 3×3: 9 multiplications
- 4×4: ~40 multiplications
- n×n: ~n! multiplications (naive)
For large matrices, LU decomposition is more efficient.
Properties of Determinants
Key properties that simplify calculations.
Multiplicative Property
det(AB) = det(A) × det(B)
The determinant of a product equals the product of determinants.
Inverse Relationship
det(A⁻¹) = 1/det(A)
Only exists when det(A) ≠ 0.
Transpose
det(Aᵀ) = det(A)
Transposing doesn't change the determinant.
Scalar Multiplication
det(kA) = kⁿ × det(A)
For n×n matrix, scalar factors out n times.
Row Operations
- Swapping rows: Changes sign of det
- Multiplying row by k: Multiplies det by k
- Adding multiple of one row to another: No change
Special Matrices
- Identity matrix: det(I) = 1
- Diagonal matrix: det = product of diagonal elements
- Triangular matrix: det = product of diagonal elements
- Orthogonal matrix: det = ±1
Applications: Cramer's Rule
Using determinants to solve linear systems.
The Method
For system Ax = b, where A is n×n:
xᵢ = det(Aᵢ)/det(A)
Aᵢ is A with column i replaced by b.
Example: 2×2 System
3x + 2y = 7 x + 4y = 9
A = |3 2| b = |7| |1 4| |9|
det(A) = 12 - 2 = 10
A₁ = |7 2| det(A₁) = 28 - 18 = 10 |9 4|
A₂ = |3 7| det(A₂) = 27 - 7 = 20 |1 9|
x = 10/10 = 1 y = 20/10 = 2
When to Use Cramer's Rule
- Small systems (2×2, 3×3)
- Need to solve for one variable only
- Theoretical/algebraic solutions
For larger systems, Gaussian elimination is more efficient.
Geometric Interpretation
Determinants measure transformation effects.
2D: Area Scaling
|det(A)| = factor by which areas are scaled
If det = 2, areas double. If det = 0.5, areas halve.
3D: Volume Scaling
|det(A)| = factor by which volumes are scaled
For rotation matrices: det = 1 (volume preserved).
Orientation
- det > 0: Preserves orientation (right-hand rule)
- det < 0: Reverses orientation (reflection)
Examples
Scaling matrix: |2 0| |0 3| det = 6 (scales x by 2, y by 3)
Rotation matrix: |cos θ -sin θ| |sin θ cos θ| det = 1 (preserves area)
Reflection: |1 0| |0 -1| det = -1 (flips over x-axis)
Shear: |1 k| |0 1| det = 1 (area preserved, shape changed)
Pro Tips
- 💡det = 0 means the matrix is singular (not invertible).
- 💡For 2×2: det = ad - bc (diagonal products).
- 💡Use cofactor expansion along rows/columns with zeros.
- 💡det(AB) = det(A) × det(B).
- 💡|det| gives the area/volume scaling factor.
- 💡Negative det means orientation is reversed.
- 💡Swapping two rows changes the sign of det.
- 💡Adding a multiple of one row to another preserves det.
- 💡Triangular matrix det = product of diagonal elements.
- 💡det(A⁻¹) = 1/det(A) when A is invertible.
- 💡det(Aᵀ) = det(A) always.
- 💡For n×n matrix: det(kA) = k^n × det(A).
Frequently Asked Questions
A zero determinant (det = 0) means the matrix is "singular" or not invertible. Geometrically, it collapses space to a lower dimension (e.g., 2D to a line). For systems of equations, det = 0 means either no solution or infinitely many solutions exist.
