1. What is a Cofactor?

The cofactor of an element in a matrix is the signed minor of that element. It is used in calculating determinants, matrix inverses, and solving systems of linear equations.

2. How Does the Calculator Work?

The calculator uses the cofactor formula:

Where:

• \( C_{ij} \) — Cofactor of element at row i, column j
• \( (-1)^{i+j} \) — Sign factor (positive if i+j is even, negative if odd)
• \( M_{ij} \) — Minor matrix obtained by removing row i and column j
• \( \det(M_{ij}) \) — Determinant of the minor matrix

Explanation: The cofactor combines the algebraic sign with the determinant of the minor matrix to provide a signed value used in various matrix operations.

3. Importance of Cofactors

Details: Cofactors are essential for calculating matrix determinants using cofactor expansion, finding matrix inverses via the adjugate matrix, and solving linear systems using Cramer's rule.

4. Using the Calculator

Tips: Enter matrix elements, select the size, and specify the row and column indices (starting from 1). The calculator will compute the cofactor and display the corresponding minor matrix.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between minor and cofactor?
A: The minor is the determinant of the submatrix, while the cofactor is the signed minor (multiplied by (-1)^(i+j)).

Q2: How are cofactors used in matrix inversion?
A: The inverse matrix is calculated as (1/det(A)) × adj(A), where adj(A) is the adjugate matrix formed from cofactors.

Q3: What does the sign factor (-1)^(i+j) represent?
A: It creates a checkerboard pattern of signs across the matrix, essential for proper determinant calculation.

Q4: Can cofactors be calculated for non-square matrices?
A: No, cofactors are only defined for square matrices since they require determinant calculation.

Q5: What is the relationship between cofactors and determinants?
A: The determinant can be computed by cofactor expansion along any row or column: det(A) = Σ a_ij × C_ij for any fixed i or j.