MathematicsChapter 2: Matrices and Determinants
Cramer, Inverse Method, Gauss
For 3 equations, 3 unknowns, det A != 0: unique solution.
Learning objectives
- State the main ideas and notation used in cramer, inverse method, gauss.
- Apply the rules of cramer, inverse method, gauss in a clear, step-by-step solution.
- Check signs, conditions, and the final answer for mathematical accuracy.
Definition
For 3 equations, 3 unknowns, det A != 0: unique solution. Inverse method: x = A-1 b. Cramer's rule: xi = det Ai / det A, where A_i replaces column i by b. Gauss elimination: row-reduce [A|b] to triangle, then back-substitute. Homogeneous 3x3: Gauss on [A|0]; if det=0, infinite (a line or plane of solutions in school language). Meaning: three named tools, same unique answer when det != 0.
Mathematical form
A = [[1,1],[2,3]], b = [[5],[8]]
Ax = [[5,1],[8,3]], det = 7
x = 71 = 7
x = A
Where / symbols
- CRAMER, INVERSE METHOD, GAUSS
- For 3 equations, 3 unknowns, det A != 0: unique solution. Inverse method: x = A^{-1} b. Cramer's rule: x_i = det A_i / det A, where A_i replaces column i by b. Gauss elimination: row-reduce [A|b] to triangle, then back-substitute. Homogeneous 3x3: Gauss on [A|0]; if det=0, infinite (a line or plane of solutions in school language). Meaning: three named tools, same unique answer when det != 0.
- 1. CRAMER AND INVERSE
- Cramer: four 3x3 dets (A and three A_i). Inverse: find A^{-1} then multiply b. Good when the paper already wants det practice.
- 2. GAUSS
- Forward eliminate. Back substitute. Fewer arithmetic slips for many students. Homogeneous: the 0 column stays 0. Show each row op.
Relationship
- x = A-1 b
Overview
Pick a method:
Cramer and Inverse
Cramer: four 3x3 dets (A and three Ai). Inverse: find A-1 then multiply b. Good when the paper already wants det practice.
Examples Replace one column · x = A^{-1} b · Check one equation
Gauss
Forward eliminate. Back substitute. Fewer arithmetic slips for many students. Homogeneous: the 0 column stays 0. Show each row op.
Examples R2 := R2 - k R1 · Do not skip a swap if pivot 0 · Free var if a missing pivot
WHAT — CRAMER, INVERSE METHOD, GAUSS
For 3 equations, 3 unknowns, det A != 0: unique solution. Inverse method: x = A-1 b. Cramer's rule: xi = det Ai / det A, where A_i replaces column i by b. Gauss elimination: row-reduce [A|b] to triangle, then back-substitute. Homogeneous 3x3: Gauss on [A|0]; if det=0, infinite (a line or plane of solutions in school language). Meaning: three named tools, same unique answer when det != 0.
RULE — 1. CRAMER AND INVERSE
Cramer: four 3x3 dets (A and three Ai). Inverse: find A-1 then multiply b. Good when the paper already wants det practice.
METHOD — 2. GAUSS
Forward eliminate. Back substitute. Fewer arithmetic slips for many students. Homogeneous: the 0 column stays 0. Show each row op.
WORKED EXAMPLE
Worked Cramer idea for x in 2x2. A = [[1,1],[2,3]], b = [[5],[8]] det A = 1 Ax = [[5,1],[8,3]], det = 7 x = 71 = 7
How to answer a cramer, inverse method, gauss 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 Cramer idea for x in 2x2.
- A = [[1,1],[2,3]], b = [[5],[8]]
- det A = 1
- Ax = [[5,1],[8,3]], det = 7
- x = 71 = 7
In simple words
Cramer needs det A != 0. If det=0, switch to Gauss to see infinite or none.
Key points
- x = A-1 b
- Cramer: det A_i / det A
- det A != 0
- triangle then back
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Cramer, Inverse Method, Gauss
Sticky Notes · Quick Revision
Core Idea
- For 3 equations, 3 unknowns, det A != 0: unique solution.
- Inverse method: x = A-1 b.
- Cramer's rule: xi = det Ai / det A, where A_i replaces column i by b.
1. CRAMER AND INVERSE
- Cramer: four 3x3 dets (A and three A_i).
- Inverse: find A-1 then multiply b.
- Good when the paper already wants det practice.
2. GAUSS
- Forward eliminate.
- Back substitute.
- Fewer arithmetic slips for many students.
- Homogeneous: the 0 column stays 0.
Worked Example
- Worked Cramer idea for x in 2x2.
- A = [[1,1],[2,3]], b = [[5],[8]]
- det A = 1
- Ax = [[5,1],[8,3]], det = 7
- x = 71 = 7
Common Misconception
- Cramer needs det A != 0. If det=0, switch to Gauss to see infinite or none.
FBISE Exam Check
- State the rule for cramer, inverse method, gauss before substituting values.
- Show the main algebra steps, keep exact values where possible, and check every condition.
60-second revision
- For 3 equations, 3 unknowns, det A != 0: unique solution.
- Inverse method: x = A-1 b.
- Cramer's rule: xi = det Ai / det A, where A_i replaces column i by b.
- Cramer needs det A != 0. If det=0, switch to Gauss to see infinite or none.
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