1. What is Matrix Cofactor Expansion?

The cofactor expansion (or Laplace expansion) is a method for computing the determinant of a square matrix by expanding along a row or column. It expresses the determinant in terms of smaller determinants called minors.

2. How Does the Calculator Work?

The calculator uses the cofactor expansion formula:

Where:

• \( A \) — Square matrix
• \( a_{1j} \) — Element in first row, j-th column
• \( C_{1j} \) — Cofactor of element \( a_{1j} \)
• \( n \) — Size of the matrix

Explanation: The method recursively breaks down the matrix into smaller submatrices until reaching 2×2 matrices whose determinants can be directly computed.

3. Importance of Determinant Calculation

Details: Determinants are fundamental in linear algebra, used for solving systems of equations, finding inverses of matrices, determining linear independence, and in transformations.

4. Using the Calculator

Tips: Enter matrix elements separated by commas for columns and semicolons for rows. For example, "1,2,3;4,5,6;7,8,9" represents a 3×3 matrix.

5. Frequently Asked Questions (FAQ)

Q1: What is the time complexity of cofactor expansion?
A: Cofactor expansion has O(n!) time complexity, making it impractical for large matrices. Other methods like LU decomposition are preferred for large matrices.

Q2: Can I expand along any row or column?
A: Yes, you can expand along any row or column, but the first row is commonly used for simplicity.

Q3: What is the relationship between determinant and matrix invertibility?
A: A matrix is invertible if and only if its determinant is non-zero.

Q4: What does a zero determinant indicate?
A: A zero determinant means the matrix is singular (not invertible) and its columns/rows are linearly dependent.

Q5: Are there more efficient methods for large matrices?
A: Yes, methods like LU decomposition or using row operations to triangular form are more efficient for large matrices.