Cofactor Expansion Calculator
Calculate matrix determinants using cofactor expansion with step-by-step solutions
Matrix Input & Configuration
Row 1 is highlighted for expansion
Quick Examples
Determinant Result
Step-by-Step Calculation
Expanding along row 1:
Element a₍1,1₎: 1
Sign factor: (-1)^2 = 1
Minor determinant: -3
Cofactor: 1 × -3 = -3
Contribution: 1 × -3 = -3
Element a₍1,2₎: 2
Sign factor: (-1)^3 = -1
Minor determinant: -6
Cofactor: -1 × -6 = 6
Contribution: 2 × 6 = 12
Element a₍1,3₎: 3
Sign factor: (-1)^4 = 1
Minor determinant: -3
Cofactor: 1 × -3 = -3
Contribution: 3 × -3 = -9
Final Result: det(A) = 1 × -3 + 2 × 6 + 3 × -3 = 0
Matrix Properties
Cofactor Formula
Cij = (-1)i+j × Mij
Cofactor = Sign × Minor
det(A) = Σ aij × Cij
Determinant = Σ Element × Cofactor
Expansion Tips
Choose rows/columns with the most zeros
Sign pattern alternates: +, -, +, -, ...
Any row or column gives the same result
For 2×2: ad - bc
Common Examples
2×2 Matrix
[2 4]
det = 3×4 - 1×2 = 10
Identity Matrix
[0 1]
det = 1 (always)
Understanding Cofactor Expansion
What is Cofactor Expansion?
Cofactor expansion is a method for calculating the determinant of a square matrix. It reduces the problem to computing several smaller determinants, making it particularly useful for matrices larger than 2×2.
Key Concepts
- •Minor: Determinant of submatrix after removing one row and column
- •Cofactor: Minor multiplied by sign factor (-1)^(i+j)
- •Expansion: Sum of element × cofactor products
Why Use Cofactor Expansion?
- •Works for any size square matrix
- •Provides step-by-step calculation method
- •Essential for understanding matrix theory
- •Used in finding matrix inverses
Tip: Always expand along the row or column with the most zeros to minimize calculations!
Mathematical Formulation
Row Expansion (along row i)
det(A) = Σ aij × Cij
Sum over all columns j in row i
Column Expansion (along column j)
det(A) = Σ aij × Cij
Sum over all rows i in column j
where Cij = (-1)i+j × Mij
Mij is the minor (determinant of submatrix after removing row i and column j)