1. What Are Minors?
The minor of an element in a matrix is the determinant of the smaller matrix formed after removing the element’s row and column.
If \( A = [a_{ij}] \), then the minor of \( a_{ij} \) is written as:
M_{ij}
To find \( M_{ij} \):
- Delete the \( i \)-th row
- Delete the \( j \)-th column
- Take the determinant of the remaining submatrix
2. Example: Finding Minors
Let:
A = \begin{bmatrix} 2 & 3 & 1 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix}
Minor of element \( a_{12} = 3 \): remove row 1 and column 2.
M_{12} = \begin{vmatrix} 4 & 6 \\ 7 & 9 \end{vmatrix} = (4)(9) - (6)(7) = 36 - 42 = -6
3. What Are Cofactors?
A cofactor includes the minor along with a sign factor. The cofactor of \( a_{ij} \) is written as \( C_{ij} \) and defined as:
C_{ij} = (-1)^{i+j} M_{ij}
This means cofactors follow a checkerboard sign pattern.
4. Sign Pattern for Cofactors
For a 3×3 matrix, the cofactor sign chart looks like:
\begin{bmatrix} + & - & + \\ - & + & - \\ + & - & + \end{bmatrix}
You multiply the minor by + or – depending on the position \( (i, j) \).
5. Example: Finding Cofactors
Using the same matrix:
A = \begin{bmatrix} 2 & 3 & 1 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix}
From earlier, we found:
M_{12} = -6
Since \( (1+2) = 3 \) which is odd:
C_{12} = (-1)^3 M_{12} = -(-6) = 6
6. Cofactor Matrix
The cofactor matrix of a matrix \( A \) is formed by arranging all cofactors \( C_{ij} \) in their corresponding positions.
This matrix is later used to find the adjoint and inverse of a matrix.
6.1. Example: Cofactor Matrix of a 2×2 Matrix
Let:
A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}
Cofactors:
- \( C_{11} = d \)
- \( C_{12} = -c \)
- \( C_{21} = -b \)
- \( C_{22} = a \)
Cofactor matrix:
\begin{bmatrix} d & -c \\ -b & a \end{bmatrix}
7. Expansion of Determinant Using Cofactors
A determinant of an \( n \times n \) matrix can be expanded along any row or column using cofactors.
If expanding along the first row:
|A| = a_{11}C_{11} + a_{12}C_{12} + a_{13}C_{13} + \cdots + a_{1n}C_{1n}
7.1. Example
Let:
A = \begin{bmatrix} 1 & 2 & 3 \\ 0 & 4 & 5 \\ 1 & -2 & 2 \end{bmatrix}
Expanding along the first row:
|A| = 1 \cdot C_{11} + 2 \cdot C_{12} + 3 \cdot C_{13}
You may compute each cofactor separately to get the final value.