1. What is a Cofactor Matrix?

The cofactor matrix is a matrix where each element is the cofactor of the corresponding element of the original matrix. The cofactor \( C_{ij} \) of element \( a_{ij} \) is \( (-1)^{i+j} \) times the minor \( M_{ij} \), which is the determinant of the submatrix formed by deleting the i-th row and j-th column.

2. How Does the Calculator Work?

The calculator uses the cofactor matrix formula:

Where:

• \( C_{ij} \) — Cofactor at position (i,j)
• \( M_{ij} \) — Minor at position (i,j)
• \( i,j \) — Row and column indices (1-based)

Explanation: For each element in the matrix, we calculate its cofactor by determining the minor (determinant of the submatrix) and multiplying by \( (-1)^{i+j} \).

3. Importance of Cofactor Matrix

Details: The cofactor matrix is essential for calculating the adjugate matrix and matrix inverse. It's used in various applications including solving systems of linear equations, computer graphics, and physics simulations.

4. Using the Calculator

Tips: Enter your square matrix using comma-separated values for columns and semicolon-separated values for rows. For example: "1,2,3;4,5,6;7,8,9" for a 3×3 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 minor multiplied by \( (-1)^{i+j} \).

Q2: Does the matrix need to be square?
A: Yes, cofactor matrices are only defined for square matrices.

Q3: What's the relationship between cofactor matrix and adjugate?
A: The adjugate is the transpose of the cofactor matrix.

Q4: Can I use this for large matrices?
A: While the calculator works for any size, computation time grows factorially with matrix size.

Q5: How is this used in matrix inversion?
A: The inverse is the adjugate divided by the determinant: \( A^{-1} = \text{adj}(A)/\det(A) \).