Cofactor Matrix Calculator
Calculate cofactor matrix, minors, and adjugate matrix with detailed step-by-step solutions
Matrix Input & Configuration
Quick Examples
Cofactor Matrix Results
Cofactor Matrix C
Adjugate Matrix (CT)
Sample Calculation: C(1,1)
Position: (1, 1)
Sign factor: (-1)^(1+1) = 1
Minor M(1,1): Remove row 1, column 1
Minor determinant: -24
Cofactor C(1,1): (-1)^{1+1} × -24 = 1 × -24 = -24
Matrix Properties
Sign Pattern
(-1)i+j
Sign Factor Formula
Pattern for 3×3
Key Formulas
Cij = (-1)i+j × Mij
Cofactor Definition
adj(A) = CT
Adjugate Matrix
A⁻¹ = adj(A) / det(A)
Matrix Inverse
Calculation Steps
Remove row i and column j to get minor Mij
Calculate determinant of the minor
Apply sign factor (-1)i+j
Multiply to get cofactor Cij
Repeat for all positions
Understanding Cofactor Matrix
What is a Cofactor Matrix?
The cofactor matrix (also called the matrix of cofactors) is a square matrix whose elements are the cofactors of the corresponding elements in the original matrix. Each cofactor is calculated by finding the minor (determinant of a submatrix) and applying the appropriate sign.
Key Applications
- •Finding matrix inverse: A⁻¹ = adj(A) / det(A)
- •Computing adjugate matrix: adj(A) = CT
- •Solving systems of linear equations
- •Understanding matrix properties and transformations
Step-by-Step Process
Step 1: Calculate Minor
For position (i,j), remove row i and column j from the original matrix to get the minor Mij.
Step 2: Find Determinant
Calculate the determinant of the minor matrix using appropriate methods (2×2 formula, cofactor expansion, etc.).
Step 3: Apply Sign
Multiply by the sign factor (-1)i+j to get the cofactor Cij.
Special Case: 2×2 Matrix
For a 2×2 matrix:
Original Matrix
[c d]
Cofactor Matrix
[-b a]
Notice the pattern: diagonal elements swap positions, anti-diagonal elements change signs.