1. What is a Minor in Matrix Algebra?

The minor of a matrix entry is the determinant of the submatrix formed by deleting the row and column containing that entry. Minors are fundamental in calculating cofactors, adjugate matrices, and determinants of larger matrices.

2. How Does the Minor Calculator Work?

The calculator uses the following process:

Where:

• \( M_{ij} \) — Minor of the element at row i, column j
• \( \text{det} \) — Determinant function
• \( i \) — Row index to remove (1-based)
• \( j \) — Column index to remove (1-based)

Explanation: The calculator first creates the submatrix by removing the specified row and column, then calculates its determinant.

3. Importance of Minors in Linear Algebra

Details: Minors are essential for computing matrix inverses, characteristic polynomials, and are used in various matrix decomposition methods. They form the basis for calculating cofactors in the adjugate matrix.

4. Using the Calculator

Tips: Enter the matrix with rows separated by semicolons and columns separated by commas. For example, "1,2,3;4,5,6;7,8,9" represents a 3×3 matrix. Specify which row and column to remove (1-based index).

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between a minor and a cofactor?
A: A cofactor is a signed minor, calculated as \( C_{ij} = (-1)^{i+j}M_{ij} \).

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

Q3: How are minors used in finding the inverse matrix?
A: The adjugate matrix (needed for inverse) is the transpose of the cofactor matrix, which is built from minors.

Q4: What's the largest matrix size this calculator supports?
A: The calculator can handle any size matrix, but determinant calculation is currently implemented only for 2×2 submatrices.

Q5: Are minors used in anything besides matrix algebra?
A: Yes, they appear in multivariate calculus (Jacobian determinants) and differential geometry.