Cofactor of a Matrix [ English ]

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1. Introduction

In matrix theory, the concept of a cofactor is fundamental for computing:

• Determinants
• Adjoint (adjugate) of a matrix
• Inverse of a matrix

Cofactors are derived from minors and incorporate a sign pattern that is essential in determinant expansion.

2. Definition of Cofactor

The cofactor of an element in a matrix is defined as:

Cij = (-1)^(i+j) × Mij

Where:

• Cij = cofactor of the element in row i, column j
• Mij = minor of that element
• (-1)^(i+j) determines the sign

3. Minor of an Element

Before finding a cofactor, we compute the minor.

Definition:

The minor (Mij) of an element is the determinant of the matrix obtained by deleting:

• the i-th row
• the j-th column

4. Sign Pattern of Cofactors

The sign of each cofactor follows a checkerboard pattern:

|  +   -   + |
|  -   +   - |
|  +   -   + |

This pattern comes from (-1)^(i+j).

5. Example: Cofactors of a 2×2 Matrix

Let:

A = | a  b |
    | c  d |

Cofactors:

• C11 = +d
• C12 = -c
• C21 = -b
• C22 = +a

Cofactor matrix:

|  d   -c |
| -b    a |

6. Example: Cofactors of a 3×3 Matrix

Let:

A = | 1  2  3 |
    | 0  4  5 |
    | 1  0  6 |

Step 1: Find Minors and Cofactors

First Row:

• C11 = +(4×6 − 5×0) = 24
• C12 = −(0×6 − 5×1) = -(-5) = 5
• C13 = +(0×0 − 4×1) = -4

Second Row:

• C21 = −(2×6 − 3×0) = -12
• C22 = +(1×6 − 3×1) = 3
• C23 = −(1×0 − 2×1) = 2

Third Row:

• C31 = +(2×5 − 3×4) = -2
• C32 = −(1×5 − 3×0) = -5
• C33 = +(1×4 − 2×0) = 4

Cofactor Matrix:

|  24   5   -4 |
| -12    3    2 |
|  -2   -5    4 |

7. Cofactor Expansion (Determinant)

Cofactors are used to compute determinants:

det(A) = a11C11 + a12C12 + a13C13

This is called expansion along a row.

8. Relation with Adjoint

The adjoint of a matrix is obtained by:

adj(A) = transpose of cofactor matrix

9. Applications of Cofactor

(a) Determinants

(b) Matrix Inverse

(c) Linear Algebra Computations

10. Key Observations

• Cofactor depends on both minor and sign
• Sign pattern is essential and must not be ignored
• Cofactors are building blocks of adjoint and inverse

11. Summary

• Cofactor is defined as: Cij = (-1)^(i+j) × Mij

• Minor is determinant after removing row and column

• Sign follows checkerboard pattern

• Used in:

  • Determinants
  • Adjoint
  • Matrix inverse