1. What is a Cofactor Matrix?

The cofactor matrix is a matrix where each element is the cofactor of the corresponding element in the original matrix. Cofactors are used in calculating the inverse of a matrix and the determinant of a matrix.

2. How Does the Calculator Work?

The calculator uses the cofactor formula:

Where:

• \( C_{ij} \) — Cofactor at row i, column j
• \( M_{ij} \) — Minor matrix obtained by removing row i and column j
• \( \det \) — Determinant of the minor matrix

Explanation: For each element in the matrix, we calculate the determinant of the submatrix formed by deleting the current row and column, then apply the sign based on the element's position.

3. Importance of Cofactor Matrix

Details: The cofactor matrix is essential for finding the adjugate matrix, which is used to compute the inverse of a matrix. It's also used in calculating determinants for larger matrices.

4. Using the Calculator

Tips: Enter the size of your square matrix (2x2 to 6x6), then fill in all the matrix elements. The calculator will compute the cofactor for each element.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between a minor and a cofactor?
A: A minor is the determinant of the submatrix, while a cofactor is the minor multiplied by (-1)^i+j where i,j are the row and column indices.

Q2: How is the cofactor matrix related to the adjugate matrix?
A: The adjugate matrix is the transpose of the cofactor matrix.

Q3: What matrices have cofactor matrices?
A: Only square matrices have cofactor matrices. Rectangular matrices don't have cofactors.

Q4: Why are cofactors important in matrix inversion?
A: The inverse of a matrix A is (1/det(A)) * adj(A), where adj(A) is the adjugate (transpose of cofactor matrix).

Q5: What's the computational complexity of calculating cofactors?
A: For an n×n matrix, it's O(n!) because it requires calculating many determinants recursively.