How to use Co-factor Expansion along the Column in Finding Determinant of a 3x3 Matrix

Calculation of 3✖3 matrix determinant by Co-factor Expansion along the Column

The Co-Factor Expansion

Definition: If A is a square matrix then the minor of Aij, denoted by Mij, which is the determinant of the sub-matrix that results from removing (deleting) the row ith and jth column of  A.

Sign Chart for a 3×3 determinant

                                                    

A sign chart is either a + or - for each element in the matrix. The first element (row 1, column 1) is always a + and it alternates from there.

Note: The + does not mean positive and the - negative. The + means the same sign as the minor and the - means the opposite of the minor. Think of it as addition and subtraction rather than positive or negative.

About the Method

To calculate a determinant you need to do the following steps:

 

1. Pick any column of the matrix

2. Find all the cofactors for that column

3. Multiply each cofactor by its matrix entry

4. Add all the values you've gotten.

 

The diagram below shows how the determinant of the third order expansion by the elements of whatever column you choose:

Note: It does not matter which column you use, the answer will be the same for any column.


This method is similar to Co-Factor Expansion along the Row.

Example 1. Find the determinant of matrix A.

Determinant by the First Column

 

Solution:

Determinant by the Second Column

 

Solution:

Determinant by the Third Row:

 

Solution:

Example 2. Find the determinant of matrix B.

Determinant by the First Column:

 

Solution:

Determinant by the Second Column:

 

Solution:

Determinant by the Third Column:

 

Solution: