Determinant of Cofactor Matrix Calculator

Matrix Determinant Formula:

\[ \det(C) = \sum_{\sigma \in S_n} \text{sgn}(\sigma) \prod_{i=1}^n C_{i,\sigma(i)} \]

Determinant of Cofactor Matrix:

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1. What is the Cofactor Matrix?

The cofactor matrix of a square matrix is obtained by replacing each element with its corresponding cofactor (the signed minor of that element). The determinant of the cofactor matrix has important mathematical properties related to the original matrix.

2. How to Calculate Determinant of Cofactor Matrix

The calculation involves two main steps:

\[ \text{1. Construct cofactor matrix } C_{ij} = (-1)^{i+j} M_{ij} \] \[ \text{2. Calculate } \det(C) \text{ using standard determinant methods} \]

Where:

\( M_{ij} \) — Minor of element \( a_{ij} \) (determinant of submatrix)
\( C_{ij} \) — Cofactor of element \( a_{ij} \)
\( \det(C) \) — Determinant of the cofactor matrix

3. Mathematical Significance

Properties: For an n×n matrix A, det(C) = det(A)^(n-1). This relationship is particularly useful in matrix theory and applications like solving systems of linear equations.

4. Using the Calculator

Instructions: Enter your matrix using comma-separated values for each row and semicolons between 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 relationship between det(C) and det(A)?
A: For an n×n matrix A, det(C) = det(A)^(n-1), where C is the cofactor matrix of A.

Q2: Can I use this for non-square matrices?
A: No, cofactor matrices and their determinants are only defined for square matrices.

Q3: What's the computational complexity?
A: Calculating det(C) is O(n!) due to the recursive nature of determinant calculation.

Q4: Are there any special cases?
A: For singular matrices (det(A)=0), det(C) will also be 0. For 1×1 matrices, det(C)=1 always.

Q5: How does this relate to the adjugate matrix?
A: The adjugate is the transpose of the cofactor matrix, so they have the same determinant.