6 min read
3x3 Cofactors
Signs + - + on the first row. Pick a row with zeros if you can. Minor = leftover 2x2. Sum a{ij} C{ij} along the chosen row.
Choose Your Level
3x3 Cofactors
What is it?
Signs + - + on the first row. Pick a row with zeros if you can. Minor = leftover 2x2. Sum a{ij} C{ij} along the chosen row.
In simple words
- Signs + - + on the first row.
- Expand row 1
- Zero entries skip work
- State the rule, show the working, and check the answer.
Important terms
3X3 COFACTORS: Signs + - + on the first row. Pick a row with zeros if you can. Minor = leftover 2x2. Sum a{ij} C{ij} along the chosen row.
3x3 COFACTORS: Signs + - + on the first row.
What you should learn
- State the rule or meaning of 3x3 cofactors in easy words.
- Apply 3x3 cofactors in a clear step-by-step solution.
- Check the conditions, working, and final answer for accuracy.
Examples
- Expand row 1
- Zero entries skip work
- Same value any row/col
MAIN IDEA
Signs + - + on the first row. Pick a row with zeros if you can. Minor = leftover 2x2. Sum a{ij} C{ij} along the chosen row.
EXAM USE
Write the rule first: Signs + - + on the first row. Substitute carefully, show one step per line, and check signs, restrictions, units, and the final result.
Recommended for Class 11
Full school-level explanation
Signs + - + on the first row. Pick a row with zeros if you can. Minor = leftover 2x2. Sum a{ij} C{ij} along the chosen row.
Understand the idea, then apply it:
WHAT — 3X3 COFACTORS
Signs + - + on the first row. Pick a row with zeros if you can. Minor = leftover 2x2. Sum a{ij} C{ij} along the chosen row.
WHY — Why it matters
This rule is a separate learning unit inside determinants 2x2 and 3x3. It can be defined, applied, and checked in its own FBISE question.
HOW — How to answer
Write the rule first: Signs + - + on the first row. Substitute carefully, show one step per line, and check signs, restrictions, units, and the final result.
MAIN IDEA
Signs + - + on the first row. Pick a row with zeros if you can. Minor = leftover 2x2. Sum a{ij} C{ij} along the chosen row.
Remember
- (-1)i+j
Examples
- Expand row 1
- Zero entries skip work
- Same value any row/col
EXAM USE
Write the rule first: Signs + - + on the first row. Substitute carefully, show one step per line, and check signs, restrictions, units, and the final result.
Remember
- focused answer
- accurate support
Examples
- State the central idea
- Use one accurate detail
- Check the final response
How to answer a 3x3 cofactors question
How it works
- Read the exact command word and identify what the question asks.
- State the matching idea, rule, or meaning before adding detail.
- Use the unit evidence or mathematical working that directly supports the answer.
- Check accuracy, relevance, and answer length before finishing.
Remember
- (-1)i+j
- 2x2: ad-bc
- 3x3: cofactor expansion
- chessboard signs
- (-1)i+j
- focused answer
- accurate support
- (-1)i+j
Deeper understanding
Go beyond memorising 3x3 Cofactors. Connect the definition to reasons, relationships, exceptions, and exam wording.
Common mistakes
Exam focus
Core Idea
What to Know
Example or Evidence
How to Answer
Exam focus
Quick revision
- (-1)i+j
- 2x2: ad-bc
- 3x3: cofactor expansion
- chessboard signs
- 3X3 COFACTORS: Signs + - + on the first row. Pick a row with zeros if you can. Minor = leftover 2x2. Sum a{ij} C{
- 3x3 COFACTORS: Signs + - + on the first row.
The sign is (-1)i+j, not 'always plus'. Chessboard of + and -.
For long answers on 3x3 Cofactors: define clearly → explain the main parts → give one example → state one exam caution.
Exam Focus
Exam Focus
Important points
- (-1)i+j
- 2x2: ad-bc
- 3x3: cofactor expansion
- chessboard signs
Important formulas
- (-1)i+j
- 2x2: ad-bc
- 3x3: cofactor expansion
Common exam traps
- The sign is (-1)i+j, not 'always plus'. Chessboard of + and -.
- Do not choose a formula before checking that its conditions fit the question.
Long question structure
Definition → key parts → example → exam caution
Common Mistakes
Common Mistakes
- Mistake
The sign is (-1)i+j, not 'always plus'. Chessboard of + and -.
- Mistake
Do not choose a formula before checking that its conditions fit the question.
- Mistake
Do not hide sign changes, restrictions, or skipped algebra steps.
- Mistake
The sign is (-1)i+j, not 'always plus'. Chessboard of + and -.
Revision Zone
60-Second Revision
Quick cues from this topic — use them for last-minute revision.
(-1)i+j
2x2: ad-bc
3x3: cofactor expansion
chessboard signs
Signs + - + on the first row. Pick a row with zeros if you can. Minor = leftover 2x2. Sum a{ij} C{ij} along the chosen row.
- Definition
- Signs + - + on the first row. Pick a row with zeros if you can. Minor = leftover 2x2. Sum a{ij} C{ij} along the chosen row.
- Example
- Expand row 1
- Critical warning
- The sign is (-1)i+j, not 'always plus'. Chessboard of + and -.
Did you understand this topic?
Your answer is saved with your learning record. It does not mark the topic as mastered.
Ready to test yourself?
You've completed the learning section.
Exam Practice
Define
Q1. Define 3x3 Cofactors.
Explain
Q2. Explain: MAIN IDEA.
Example
Q3. Give one clear example of 3x3 Cofactors and say why it fits.
Mistake
Q4. State one common mistake students make about 3x3 Cofactors.
Exam note
Q5. Write a short exam note on 3x3 Cofactors: definition, two points, one example.
Sources & References
School-standard facts for Matric and Intermediate. Original explanations — not copied from textbooks.