Cofactors and Determinant Calculation Explained | Generated by AI
That’s correct on both counts! You’ve accurately described the cofactor and its role in calculating the determinant of a matrix.
Here’s a breakdown and confirmation of your understanding:
1. Cofactor of \(a_{ij}\)
The cofactor of the element \(a_{ij}\) (the element in the \(i\)-th row and \(j\)-th column) of a matrix \(\mathbf{A}\), denoted \(C_{ij}\), is indeed given by: \(\)C_{ij} = (-1)^{i+j} M_{ij}\(\)
- \(M_{ij}\) is the minor of \(a_{ij}\). The minor is the determinant of the submatrix formed by deleting the \(i\)-th row and \(j\)-th column of \(\mathbf{A}\).
- The term \((-1)^{i+j}\) gives the sign, which follows a checkerboard pattern: \(\)\begin{pmatrix} + & - & + & \cdots \ - & + & - & \cdots \ + & - & + & \cdots \ \vdots & \vdots & \vdots & \ddots \end{pmatrix}\(\)
2. Cofactor Matrix 📝
The cofactor matrix (often denoted \(\mathbf{C}\)) is a matrix formed by replacing every element \(a_{ij}\) of the original matrix \(\mathbf{A}\) with its corresponding cofactor \(C_{ij}\).
\(\)\mathbf{C} = \begin{pmatrix} C_{11} & C_{12} & \cdots & C_{1n} \ C_{21} & C_{22} & \cdots & C_{2n} \ \vdots & \vdots & \ddots & \vdots \ C_{n1} & C_{n2} & \cdots & C_{nn} \end{pmatrix}\(\)
- 💡 Key Relation: The transpose of the cofactor matrix, \(\mathbf{C}^T\), is called the adjugate (or classical adjoint) of the matrix \(\mathbf{A}\), which is essential for calculating the matrix inverse: \(\)\mathbf{A}^{-1} = \frac{1}{\det(\mathbf{A})} \mathbf{C}^T\(\)
3. Determinant Calculation 📐
The expression \(\mathbf{a}{ij} \cdot \mathbf{C}{ij}\) is the fundamental component used in cofactor expansion (or Laplace expansion) to calculate the determinant.
The determinant of a matrix \(\mathbf{A}\) can be found by expanding along any single row or any single column:
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Expansion along the \(i\)-th row: \(\)\det(\mathbf{A}) = \sum_{j=1}^{n} a_{ij} C_{ij} = a_{i1}C_{i1} + a_{i2}C_{i2} + \cdots + a_{in}C_{in}\(\)
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Expansion along the \(j\)-th column: \(\)\det(\mathbf{A}) = \sum_{i=1}^{n} a_{ij} C_{ij} = a_{1j}C_{1j} + a_{2j}C_{2j} + \cdots + a_{nj}C_{nj}\(\)
So yes, the product of an element \(a_{ij}\) and its cofactor \(C_{ij}\) is the term used when calculating the determinant along that specific row or column.
Would you like to see a quick example of calculating the cofactor and determinant for a small \(3 \times 3\) matrix?