Start with a square matrix
A =abcd
The determinant is written as det(A) or |A|.
Learning Objectives
- understand the main ideas in Determinants;
- use correct mathematical language and notation;
- apply the ideas to progressively harder problems;
- check and explain each solution.
Introduction
This lesson develops Determinants from meaning and patterns before moving to calculations. Look for what the quantities, shapes or symbols represent, then connect that idea to the method.
Key Notes
Read the definitions and key facts below carefully. Each rule is most useful when you can explain why it fits the situation, not simply recall it.
🌱 What Is a Determinant?
A determinant is a single number calculated from a square matrix. For a 2 × 2 matrix, it helps tell us whether the matrix has an inverse.
Follow the diagonals
abcd
Multiply the main diagonal: a × d. Multiply the other diagonal: b × c.
Subtract in the correct order
det(A) = ad − bc
Main diagonal product minus other diagonal product.
Example
4235
(4 × 5) − (2 × 3) = 20 − 6 = 14.
Why zero matters
If det(A) = 0, the matrix is called singular and has no inverse.
If det(A) ≠ 0, an inverse exists.
Think First: A determinant is only defined here for a square matrix. Check the order before using ad − bc.
Rule
For abcd, determinant=ad−bc.
Example
|4325|=4×5−3×2=14.
Singular vs Non-singular
Singular
det=0; no inverse.
Non-singular
det≠0; inverse exists.
Practice Questions
- Find det3254.
- Is 2412 singular?
- Find x if detx234=10.
✅ Check Your Work
Show answers and working
- 2.
- Yes; determinant=0.
- 4x−6=10, so x=4.
🔷 Determinant Explorer
📘 Remember
Key idea
For a 2×2 matrix, det(A)=ad−bc. A zero determinant means the matrix is singular.
⚠️ Common Mistakes
The subtraction order matters: main diagonal product minus the other diagonal product.
⭐ Exam Tips
Write ad−bc first, substitute carefully, and keep brackets around negative numbers.
🌍 Did You Know?
The absolute determinant measures how a transformation changes area.
Common Mistakes
- Choosing a rule before identifying what the question describes.
- Skipping working or changing notation part-way through a solution.
- Accepting an answer without checking its size, sign or units.
Exam Focus
Show the mathematical relationship you are using, keep notation consistent and give a clear final answer. Use a quick estimate, diagram or inverse operation to check that the result is reasonable.
Summary
- Understand the meaning of each idea before selecting a method.
- Use definitions, properties and notation consistently.
- Show working and check that the final result is sensible.