Calculating 2x2 Determinant

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|

1. Identify elements:

   • a = 2, b = 3
   • c = 4, d = 5

2. Main diagonal:

   • 2 × 5 = 10

3. Secondary diagonal:

   • 3 × 4 = 12

4. 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. |1 2| |3 4|

2. |5 0| |0 5|

3. |2 1| |-1 2|

Solutions are provided at the bottom of this guide.