Calculating 2x2 Determinant
Learn how to calculate the determinant of a 2x2 matrix step by step
• 1 min read
linear-algebra determinants matrices
The determinant of a 2x2 matrix is one of the fundamental calculations in linear algebra. Here’s how to do it step by step.
The Formula
For a 2x2 matrix:
|a b|
|c d|
The determinant is calculated as: det = ad - bc
Step-by-Step Process
1. Identify the Elements
First, identify the elements of your matrix:
- Top-left (a)
- Top-right (b)
- Bottom-left (c)
- Bottom-right (d)
2. Main Diagonal
Multiply the elements on the main diagonal (top-left to bottom-right):
- Result = a × d
3. Secondary Diagonal
Multiply the elements on the secondary diagonal (top-right to bottom-left):
- Result = b × c
4. Final Calculation
Subtract the secondary diagonal product from the main diagonal product:
- Determinant = (a × d) - (b × c)
Example
Let’s calculate the determinant of:
|2 3|
|4 5|
-
Identify elements:
- a = 2, b = 3
- c = 4, d = 5
-
Main diagonal:
- 2 × 5 = 10
-
Secondary diagonal:
- 3 × 4 = 12
-
Final calculation:
- det = 10 - 12 = -2
Properties to Remember
- If the determinant is zero, the matrix is singular (non-invertible)
- The determinant can be positive or negative
- The determinant represents the area scaling factor of the linear transformation
Practice Problems
Try calculating these determinants:
-
|1 2| |3 4|
-
|5 0| |0 5|
-
|2 1| |-1 2|
Solutions are provided at the bottom of this guide.