Cofactor Expansion

Overview

Cofactor expansion (Laplace expansion) is a method for computing determinants of any size matrix by reducing it to smaller determinants. It's especially useful when a row or column has many zeros.

Definitions

Minor

The minor $M_{ij}$ is the determinant of the $(n-1) \times (n-1)$ submatrix obtained by deleting row $i$ and column $j$.

Cofactor

The cofactor $C_{ij}$ is the signed minor:

$C_{ij} = (-1)^{i+j} \times M_{ij}$

Sign Pattern

$\begin{bmatrix} + & - & + & - & \cdots \\ - & + & - & + & \cdots \\ + & - & + & - & \cdots \\ - & + & - & + & \cdots \\ \vdots & \vdots & \vdots & \vdots & \ddots \end{bmatrix}$

Entry $(i,j)$ has sign $(-1)^{i+j}$.

Cofactor Expansion Formula

Along Row $i$

$\det(A) = \sum_{j=1}^{n} a_{ij} C_{ij} = a_{i1}C_{i1} + a_{i2}C_{i2} + \cdots + a_{in}C_{in}$

Along Column $j$

$\det(A) = \sum_{i=1}^{n} a_{ij} C_{ij} = a_{1j}C_{1j} + a_{2j}C_{2j} + \cdots + a_{nj}C_{nj}$

3×3 Example

$A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix}$

Expansion Along Row 1:

$\det(A) = 1 \times C_{11} + 2 \times C_{12} + 3 \times C_{13}$

Calculate minors:

$M_{11} = \begin{vmatrix} 5 & 6 \\ 8 & 9 \end{vmatrix} = 5(9) - 6(8) = -3$

$M_{12} = \begin{vmatrix} 4 & 6 \\ 7 & 9 \end{vmatrix} = 4(9) - 6(7) = -6$

$M_{13} = \begin{vmatrix} 4 & 5 \\ 7 & 8 \end{vmatrix} = 4(8) - 5(7) = -3$

Calculate cofactors:

$C_{11} = (+1)(-3) = -3$

$C_{12} = (-1)(-6) = 6$

$C_{13} = (+1)(-3) = -3$

Determinant:

$\det(A) = 1(-3) + 2(6) + 3(-3) = -3 + 12 - 9 = 0$

4×4 Example

$A = \begin{bmatrix} 1 & 0 & 0 & 3 \\ 2 & 1 & 0 & 4 \\ 0 & 0 & 5 & 6 \\ 0 & 0 & 0 & 7 \end{bmatrix}$

Expand along column 1 (most zeros):

$\det(A) = 1 \times C_{11} + 2 \times C_{21} + 0 \times C_{31} + 0 \times C_{41} = 1 \times C_{11} + 2 \times C_{21}$

Calculate $C_{11}$:

$M_{11} = \begin{vmatrix} 1 & 0 & 4 \\ 0 & 5 & 6 \\ 0 & 0 & 7 \end{vmatrix} = 1(5 \times 7 - 6 \times 0) = 35 \quad \text{(triangular: product of diagonal)}$

$C_{11} = (+1)(35) = 35$

Calculate $C_{21}$:

$M_{21} = \begin{vmatrix} 0 & 0 & 3 \\ 0 & 5 & 6 \\ 0 & 0 & 7 \end{vmatrix}$

Expand along column 1: all entries are 0, so $M_{21} = 0$

$C_{21} = (-1)(0) = 0$

Final:

$\det(A) = 1(35) + 2(0) = 35$

Strategy: Choose Best Row/Column

Always expand along the row or column with the most zeros to minimize computation.

Cofactor Matrix

The matrix of cofactors:

$C = \begin{bmatrix} C_{11} & C_{12} & \cdots & C_{1n} \\ C_{21} & C_{22} & \cdots & C_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ C_{n1} & C_{n2} & \cdots & C_{nn} \end{bmatrix}$

Adjugate (Adjoint)

The adjugate is the transpose of the cofactor matrix:

$\text{adj}(A) = C^T$

Computing Inverse via Cofactors

$A^{-1} = \frac{1}{\det(A)} \times \text{adj}(A)$

Example

For $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$:

$C = \begin{bmatrix} d & -c \\ -b & a \end{bmatrix}$

$\text{adj}(A) = \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}$

$A^{-1} = \frac{1}{ad-bc} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}$