How to Calculate a 3x3 Matrix Determinant
Master the Sarrus' rule and cofactor expansion methods for calculating 3x3 matrix determinants with clear examples.
• 1 min read
determinants 3x3-matrix sarrus-rule cofactor-expansion
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:
- Write the first two rows again below the matrix
- Multiply along the diagonals:
- Main diagonals (positive terms)
- Secondary diagonals (negative terms)
- 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:
- Choose first row for expansion
- Calculate minors for each element
- Apply alternating signs
- Multiply and sum
Example Calculation
Let’s calculate the determinant of:
|2 -1 3|
|4 5 0|
|1 2 6|
Using Cofactor Expansion:
- First term: 2[(5×6) - (0×2)] = 2(30-0) = 60
- Second term: -(-1)[(4×6) - (0×1)] = 24
- Third term: 3[(4×2) - (5×1)] = 3(8-5) = 9
Final determinant = 60 + 24 + 9 = 93
Practice Problems
Try these matrices:
-
|1 2 3| |0 1 4| |5 6 0|
-
|2 0 0| |0 2 0| |0 0 2|
-
|1 1 1| |2 2 1| |3 1 3|
Solutions provided at the bottom of the guide.