Calculating a 3x3 Matrix Determinant

This guide covers two methods for calculating the determinant of a 3x3 matrix: Sarrus’ rule and cofactor expansion.

The Matrix Structure

For a 3x3 matrix:

|a b c|
|d e f|
|g h i|

Method 1: Sarrus’ Rule

Steps:

1. Write the first two rows again below the matrix
2. Multiply along the diagonals:
   • Main diagonals (positive terms)
   • Secondary diagonals (negative terms)
3. Add all terms together

Visual Representation:

|a b c|
|d e f| → aei + bfg + cdh - ceg - bdi - afh
|g h i|
|a b c|
|d e f|

Method 2: Cofactor Expansion

The Formula

det = a(ei-fh) - b(di-fg) + c(dh-eg)

Steps:

1. Choose first row for expansion
2. Calculate minors for each element
3. Apply alternating signs
4. Multiply and sum

Example Calculation

Let’s calculate the determinant of:

|2 -1  3|
|4  5  0|
|1  2  6|

Using Cofactor Expansion:

1. First term: 2[(5×6) - (0×2)] = 2(30-0) = 60
2. Second term: -(-1)[(4×6) - (0×1)] = 24
3. Third term: 3[(4×2) - (5×1)] = 3(8-5) = 9

Final determinant = 60 + 24 + 9 = 93

Practice Problems

Try these matrices:

1. |1 2 3| |0 1 4| |5 6 0|

2. |2 0 0| |0 2 0| |0 0 2|

3. |1 1 1| |2 2 1| |3 1 3|

Solutions provided at the bottom of the guide.