Cofactor Expansion

Cofactor expansion is a method used to calculate the Determinant of an $n \times n$ matrix $A$ . It works by reducing the determinant of the $n \times n$ matrix to a sum of determinants of $(n - 1) \times (n - 1)$ submatrices. It is a recursive definition.

$C_{ij} = (- 1)^{i + j} det (M_{ij})$ where $M_{ij}$ is the determinant of the submatrix of $A$ obtained by deleting the $i$ -th row and $j$ -th column

The $\pm$ signs for each term of the expansion are determined solely by the position $(i, j)$ and follow a checkerboard pattern, starting with $+$ in the top-left corner

+ - + ⋮ - + - ⋮ + - + ⋮ \dots \dots \dots ⋱

The determinant of a matrix $A$ can be calculated by cofactor expansion along any single row $i$ or any single column $j$ .

Expansion Along Row $i$ : $det (A) = a_{i 1} C_{i 1} + a_{i 2} C_{i 2} + \dots + a_{in} C_{in}$

Expansion Along Column $j$ : $det (A) = a_{1 j} C_{1 j} + a_{2 j} C_{2 j} + \dots + a_{nj} C_{nj}$

Strategy: To simplify calculations, it is generally best to choose the row or column with the most zeros.

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Cofactor Expansion

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