Matrix of Cofactors

Matrix of cofactors

The matrix of cofactors is an important step in finding the adjoint and inverse of a square matrix. It's constructed by replacing each element of the original matrix with its corresponding cofactor.

Definition

Given a square matrix A, the cofactor of an element $a_{ij}$ (located in the i-th row and j-th column) is denoted as $C_{ij}$ and is defined as:

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

Where $M_{ij}$ is the minor of the element $a_{ij}$. The minor $M_{ij}$ is the determinant of the submatrix formed by deleting the i-th row and j-th column of A.

Steps to find the matrix of cofactors

1. Find the Minors: For each element $a_{ij}$ in the matrix A, find its minor $M_{ij}$ by deleting the i-th row and j-th column and calculating the determinant of the resulting submatrix.
2. Apply the Sign: Multiply each minor $M_{ij}$ by $(-1)^{i+j}$. This alternating sign pattern is often visualized as a checkerboard pattern:

$\begin{bmatrix} + & - & + \\ - & + & - \\ + & - & + \end{bmatrix}$

(for a 3x3 matrix). 3. Construct the Cofactor Matrix: Arrange the calculated cofactors $C_{ij}$ into a matrix of the same size as the original matrix A.

Example: finding the matrix of cofactors for a 3x3 matrix

Let's consider the matrix:

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

1. Finding the Minors:

$$M_{11} = \begin{vmatrix} 5 & 6 \\ 8 & 9 \end{vmatrix} = (5 \times 9) - (6 \times 8) = 45 - 48 = -3$$ $$M_{12} = \begin{vmatrix} 4 & 6 \\ 7 & 9 \end{vmatrix} = (4 \times 9) - (6 \times 7) = 36 - 42 = -6$$ $$M_{13} = \begin{vmatrix} 4 & 5 \\ 7 & 8 \end{vmatrix} = (4 \times 8) - (5 \times 7) = 32 - 35 = -3$$ $$M_{21} = \begin{vmatrix} 2 & 3 \\ 8 & 9 \end{vmatrix} = (2 \times 9) - (3 \times 8) = 18 - 24 = -6$$ $$M_{22} = \begin{vmatrix} 1 & 3 \\ 7 & 9 \end{vmatrix} = (1 \times 9) - (3 \times 7) = 9 - 21 = -12$$ $$M_{23} = \begin{vmatrix} 1 & 2 \\ 7 & 8 \end{vmatrix} = (1 \times 8) - (2 \times 7) = 8 - 14 = -6$$ $$M_{31} = \begin{vmatrix} 2 & 3 \\ 5 & 6 \end{vmatrix} = (2 \times 6) - (3 \times 5) = 12 - 15 = -3$$ $$M_{32} = \begin{vmatrix} 1 & 3 \\ 4 & 6 \end{vmatrix} = (1 \times 6) - (3 \times 4) = 6 - 12 = -6$$ $$M_{33} = \begin{vmatrix} 1 & 2 \\ 4 & 5 \end{vmatrix} = (1 \times 5) - (2 \times 4) = 5 - 8 = -3$$

2. Applying the Sign:

$$\begin{aligned} C_{11} &= (+1) \times M_{11} = (+1) \times (-3) = -3 \\ C_{12} &= (-1) \times M_{12} = (-1) \times (-6) = 6 \\ C_{13} &= (+1) \times M_{13} = (+1) \times (-3) = -3 \\ C_{21} &= (-1) \times M_{21} = (-1) \times (-6) = 6 \\ C_{22} &= (+1) \times M_{22} = (+1) \times (-12) = -12 \\ C_{23} &= (-1) \times M_{23} = (-1) \times (-6) = 6 \\ C_{31} &= (+1) \times M_{31} = (+1) \times (-3) = -3 \\ C_{32} &= (-1) \times M_{32} = (-1) \times (-6) = 6 \\ C_{33} &= (+1) \times M_{33} = (+1) \times (-3) = -3 \end{aligned}$$

3. Constructing the Cofactor Matrix:

$$\text{Cof}(A) = \begin{bmatrix} -3 & 6 & -3 \\ 6 & -12 & 6 \\ -3 & 6 & -3 \end{bmatrix}$$

This is the matrix of cofactors for matrix A.