Cofactor Expansion

How do you find the determinant of a 3x3 matrix without memorizing a complex formula? Cofactor expansion provides a systematic method to calculate the determinant of any square matrix.

Identifying the Minor: A Smaller Determinant

Minor

The minor

M_{ij}

of an element

a_{ij}

in a matrix is the determinant of the submatrix formed by deleting the

i

-th row and

j

-th column of the original matrix.

Example: For a matrix

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

, the minor

M_{11}

(for element 1) is the determinant of the submatrix formed by removing row 1 and column 1:

M_{11} = \det\begin{pmatrix} 5 & 6 \\ 8 & 9 \end{pmatrix} = (5 \times 9) - (6 \times 8) = 45 - 48 = -3

The Cofactor: Minor with a Sign

Cofactor

The cofactor

C_{ij}

of an element

a_{ij}

is the minor

M_{ij}

multiplied by

(-1)^{i+j}

. This term determines the sign of the minor in the expansion.

Example: Using the previous matrix

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

, we found

M_{11} = -3

. The cofactor

C_{11}

is calculated as

C_{11} = (-1)^{1+1} \times M_{11} = (-1)^2 \times (-3) = 1 \times (-3) = -3

. For

M_{12}

, if it were 5,

C_{12} = (-1)^{1+2} \times 5 = -1 \times 5 = -5

📌 Cofactor Sign Pattern

The $(-1)^{i+j}$ term creates an alternating sign pattern across the matrix, often called the 'checkerboard pattern':

\begin{pmatrix} + & - & + \\ - & + & - \\ + & - & + \end{pmatrix}

The Cofactor Expansion Formula

📐 Determinant by Cofactor Expansion

The determinant of an $n \times n$ matrix $A$ can be found by expanding along any row $i$ or any column $j$ .

Expansion along row $i$ :

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

Expansion along column $j$ :

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

Step-by-Step: Calculating a 3x3 Determinant

Calculating $\det(A)$ for $A = \begin{pmatrix} 1 & 2 & 3 \\ 0 & 4 & 5 \\ 1 & 0 & 6 \end{pmatrix}$

Choose a Row or Column

For simplicity, we will expand along the first row. The elements are

a_{11}=1

a_{12}=2

, and

a_{13}=3

Calculate

C_{11}

for

a_{11}=1

The minor

M_{11}

is the determinant of the submatrix after removing row 1 and column 1:

\det\begin{pmatrix} 4 & 5 \\ 0 & 6 \end{pmatrix} = (4 \times 6) - (5 \times 0) = 24 - 0 = 24

The cofactor

C_{11} = (-1)^{1+1} \times M_{11} = 1 \times 24 = 24

. The term for the determinant is

a_{11}C_{11} = 1 \times 24 = 24

Calculate

C_{12}

for

a_{12}=2

The minor

M_{12}

is the determinant of the submatrix after removing row 1 and column 2:

\det\begin{pmatrix} 0 & 5 \\ 1 & 6 \end{pmatrix} = (0 \times 6) - (5 \times 1) = 0 - 5 = -5

The cofactor

C_{12} = (-1)^{1+2} \times M_{12} = -1 \times (-5) = 5

. The term for the determinant is

a_{12}C_{12} = 2 \times 5 = 10

Calculate

C_{13}

for

a_{13}=3

The minor

M_{13}

is the determinant of the submatrix after removing row 1 and column 3:

\det\begin{pmatrix} 0 & 4 \\ 1 & 0 \end{pmatrix} = (0 \times 0) - (4 \times 1) = 0 - 4 = -4

The cofactor

C_{13} = (-1)^{1+3} \times M_{13} = 1 \times (-4) = -4

. The term for the determinant is

a_{13}C_{13} = 3 \times (-4) = -12

Sum the Terms

Add the calculated terms to find the determinant:

\det(A) = (a_{11}C_{11}) + (a_{12}C_{12}) + (a_{13}C_{13}) = 24 + 10 + (-12) = 34 - 12 = 22

Thus, the determinant of matrix

A

is 22.

Check Your Understanding

For a

4 \times 4

matrix, what is the sign of the cofactor

C_{23}

Cofactor Expansion in Python

pythonCalculating a 3x3 Determinant with Cofactor Expansion

Try It Yourself

Modify the determinant_3x3_cofactor function to expand along the second column instead of the first row. Verify that it produces the same determinant for matrix_A.

python

import numpy as np

def get_submatrix(matrix, row_to_remove, col_to_remove):
    return [row[:col_to_remove] + row[col_to_remove+1:]
            for i, row in enumerate(matrix) if i != row_to_remove]

def determinant_2x2(matrix):
    return matrix[0][0] * matrix[1][1] - matrix[0][1] * matrix[1][0]

def determinant_3x3_cofactor(matrix):
    if len(matrix) != 3 or len(matrix[0]) != 3:
        raise ValueError("Matrix must be 3x3")

det = 0
    # Current expansion is along the first row (i=0)
    # Modify this loop to expand along the second column (j=1)
    # Remember to adjust the row index 'i' and column index 'j' for (-1)**(i+j)
    for j in range(3):
        element = matrix[0][j]
        submatrix = get_submatrix(matrix, 0, j)
        minor = determinant_2x2(submatrix)
        cofactor = ((-1)**(0+j)) * minor 
        det += element * cofactor
    return det

matrix_A = [
    [1, 2, 3],
    [0, 4, 5],
    [1, 0, 6]
]

det_A = determinant_3x3_cofactor(matrix_A)
print(f"The determinant of matrix A (expanded along first row) is: {det_A}")

# Call the modified function here and print its result

Strategic Expansion: Choosing Your Row or Column

While cofactor expansion works for any row or column, a smart choice can significantly reduce the number of calculations. If a matrix has a row or column containing many zero elements, expanding along that row or column simplifies the process. Any term $a_{ij}C_{ij}$ where $a_{ij}=0$ will automatically be zero, eliminating the need to calculate its corresponding minor and cofactor. This strategy is particularly useful for larger matrices or those with sparse structures.

Key Takeaways

Cofactor expansion offers a systematic, recursive method to calculate the determinant of any square matrix.
A minor $M_{ij}$ is the determinant of the submatrix formed by removing row $i$ and column $j$ .
A cofactor $C_{ij}$ is the minor $M_{ij}$ multiplied by $(-1)^{i+j}$ , which follows a checkerboard sign pattern.
The determinant is the sum of products of elements and their cofactors along any chosen row or column: $\det(A) = \sum a_{ij}C_{ij}$ .
Strategically choosing a row or column with many zero elements simplifies calculations, as terms with zero elements do not contribute to the sum.

← All lessons in Determinant of a Matrix

Cofactor Expansion

Identifying the Minor: A Smaller Determinant

The Cofactor: Minor with a Sign

The Cofactor Expansion Formula

Step-by-Step: Calculating a 3x3 Determinant

Cofactor Expansion in Python

Strategic Expansion: Choosing Your Row or Column

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