1. What are Matrix Minors and Cofactors?

The minor M_ij of a matrix is the determinant of the submatrix formed by deleting the i-th row and j-th column. The cofactor C_ij is the minor multiplied by (-1)^i+j.

2. How Does the Calculator Work?

The calculator uses these formulas:

Where:

• \( M_{ij} \) — Minor of element at row i, column j
• \( C_{ij} \) — Cofactor of element at row i, column j
• \( \text{submatrix} \) — Original matrix with row i and column j removed
• \( \text{det} \) — Determinant function

3. Importance of Minors and Cofactors

Details: Minors and cofactors are essential for calculating matrix inverses, adjugates, and determinants. They are fundamental concepts in linear algebra with applications in solving systems of equations, computer graphics, and engineering.

4. Using the Calculator

Tips: Enter your matrix in the format "1,2,3;4,5,6" for a 2×3 matrix. Specify which row and column to remove (1-based index). The calculator will compute both the minor and cofactor.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between a minor and a cofactor?
A: A minor is just the determinant of the submatrix, while a cofactor includes the sign factor (-1)^i+j.

Q2: Can I calculate minors for non-square matrices?
A: No, minors are only defined for square matrices since determinants require square matrices.

Q3: How are cofactors used in matrix inversion?
A: The adjugate matrix (used in inversion) is the transpose of the cofactor matrix.

Q4: What's the time complexity for calculating minors?
A: For an n×n matrix, calculating one minor is O(n³) using Laplace expansion, same as calculating a determinant.

Q5: Are there more efficient methods for large matrices?
A: For large matrices, LU decomposition or other numerical methods are more efficient than direct minor calculation.