MathematicsChapter 2: Matrices and Determinants
Exam Response (matrices)
Typical tasks: size of AB; compute AB; det 3x3; inverse 2x2; rank; Cramer or Gauss on 3x3; consistent vs not.
Learning objectives
- State the main ideas and notation used in exam response (matrices).
- Apply the rules of exam response (matrices) in a clear, step-by-step solution.
- Check signs, conditions, and the final answer for mathematical accuracy.
Definition
Typical tasks: size of AB; compute AB; det 3x3; inverse 2x2; rank; Cramer or Gauss on 3x3; consistent vs not. Write sizes before you multiply. One entry of a product per line if needed.
Mathematical form
A = [[4,3],[2,2]]
det = 8-6 = 2
det = 8-6
Where / symbols
- EXAM RESPONSE (MATRICES)
- Typical tasks: size of AB; compute AB; det 3x3; inverse 2x2; rank; Cramer or Gauss on 3x3; consistent vs not. Write sizes before you multiply. One entry of a product per line if needed.
- 1. COMPUTE
- Add same size. Multiply row-col. det cofactors. Inverse / det. Rank = pivots.
- 2. SYSTEMS
- det != 0 unique. 0=1 none. Free var infinite. Homogeneous always 0.
Relationship
- det=0: no unique inverse solution
Overview
Compute, then interpret:
Compute
Add same size. Multiply row-col. det cofactors. Inverse / det. Rank = pivots.
Examples ad-bc · [A|I] · AB != BA
Systems
det != 0 unique. 0=1 none. Free var infinite. Homogeneous always 0.
Examples Cramer A_i · Gauss back-sub · Ax=0 trivial
WHAT — EXAM RESPONSE (MATRICES)
Typical tasks: size of AB; compute AB; det 3x3; inverse 2x2; rank; Cramer or Gauss on 3x3; consistent vs not. Write sizes before you multiply. One entry of a product per line if needed.
RULE — 1. COMPUTE
Add same size. Multiply row-col. det cofactors. Inverse / det. Rank = pivots.
METHOD — 2. SYSTEMS
det != 0 unique. 0=1 none. Free var infinite. Homogeneous always 0.
WORKED EXAMPLE
Worked 4-mark 2x2 det and inverse. A = [[4,3],[2,2]] det = 8-6 = 2 A-1 = (12) [[2,-3],[-2,4]] = [[1, -32],[-1, 2]]
How to answer a exam response (matrices) question
- Identify the given quantities, notation, and required result.
- Choose the matching rule and write it before substitution.
- Work one algebraic step at a time, keeping brackets and signs clear.
- Check restrictions, simplify the result, and verify it where possible.
Worked example
- Worked 4-mark 2x2 det and inverse.
- A = [[4,3],[2,2]]
- det = 8-6 = 2
- A-1 = (12) [[2,-3],[-2,4]]
- = [[1, -32],[-1, 2]]
In simple words
Show the product check A A-1 = I if marks allow.
Key points
- inner sizes match
- det=0: no unique inverse solution
- size then arithmetic
- last row
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Exam Response (matrices)
Sticky Notes · Quick Revision
Core Idea
- Typical tasks: size of AB; compute AB; det 3x3; inverse 2x2; rank; Cramer or Gauss on 3x3; consistent vs not.
- Write sizes before you multiply.
- One entry of a product per line if needed.
1. COMPUTE
- Add same size.
- Multiply row-col.
- det cofactors.
- Inverse / det.
2. SYSTEMS
- det != 0 unique.
- 0=1 none.
- Free var infinite.
- Homogeneous always 0.
Worked Example
- Worked 4-mark 2x2 det and inverse.
- A = [[4,3],[2,2]]
- det = 8-6 = 2
- A-1 = (12) [[2,-3],[-2,4]]
- = [[1, -32],[-1, 2]]
Common Misconception
- Show the product check A A-1 = I if marks allow.
FBISE Exam Check
- State the rule for exam response (matrices) before substituting values.
- Show the main algebra steps, keep exact values where possible, and check every condition.
60-second revision
- Typical tasks: size of AB; compute AB; det 3x3; inverse 2x2; rank; Cramer or Gauss on 3x3; consistent vs not.
- Write sizes before you multiply.
- One entry of a product per line if needed.
- Show the product check A A-1 = I if marks allow.
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