MathematicsChapter 3: Matrices and Determinants
DETERMINANT OF A 2x2 MATRIX
For A = [[a, b], [c, d]], |A| = ad - bc.
Learning objectives
- Explain determinant of a 2x2 matrix using the correct definitions, conditions, and notation.
- Apply ad - bc, 0 => singular correctly in routine and exam-style questions.
- Show each important step, simplify the result, and check that it satisfies the original conditions.
Definition
For A = [[a, b], [c, d]], |A| = ad - bc. It is a single number, not a matrix. |A|=0 means singular (no inverse). |A| != 0 means non-singular (inverse exists). A diagonal 2x2 has |A| = product of the diagonal entries.
Where / symbols
- DETERMINANT OF A 2x2 MATRIX
- For A = [[a, b], [c, d]], |A| = ad - bc. It is a single number, not a matrix. |A|=0 means singular (no inverse). |A| != 0 means non-singular (inverse exists). A diagonal 2x2 has |A| = product of the diagonal entries.
- COMPUTE
- [[3,1],[2,4]]: 3*4 - 1*2 = 10. [[1,2],[3,4]]: 4-6=-2. [[2,4],[1,2]]: 4-4=0, singular.
- WHAT |A| TELLS YOU
- Non-zero: inverse exists (next page). Zero: do not write A^{-1}. Determinant is not the same as adding the diagonal (that is the trace, not in this chapter's main list).
Relationship
- 0 => singular
Overview
Main diagonal minus the other:
Compute
[[3,1],[2,4]]: 3*4 - 1*2 = 10. [[1,2],[3,4]]: 4-6=-2. [[2,4],[1,2]]: 4-4=0, singular.
Examples 3*4-1*2=10 · 1*4-2*3=-2 · 2*2-4*1=0
What |a| Tells You
Non-zero: inverse exists (next page). Zero: do not write A-1. Determinant is not the same as adding the diagonal (that is the trace, not in this chapter's main list).
Examples |A|=10 invertible · |A|=0 no inverse · Not a+d
WHAT — DETERMINANT OF A 2x2 MATRIX
For A = [[a, b], [c, d]], |A| = ad - bc. It is a single number, not a matrix. |A|=0 means singular (no inverse). |A| != 0 means non-singular (inverse exists). A diagonal 2x2 has |A| = product of the diagonal entries. The symbols and conditions are part of the mathematics, so include them when stating the rule.
WHY — Why this topic matters
Determinant Of A 2X2 Matrix matters because matrices organise numbers and give short methods for transformations and simultaneous equations. Correct order, determinant, and multiplication rules prevent most exam errors.
HOW — How to solve and explain it
Start by identifying what is given and what must be found. Work through COMPUTE, then WHAT |A| TELLS YOU. Worked: A = [[5, -2], [3, 1]]. Find |A|. |A| = (5)(1) - (-2)(3) |A| = 5 + 6 |A| = 11 11 != 0 so A is non-singular Finish by checking signs, restrictions, units, and whether the answer matches the question.
Worked example
- A = [[5, -2], [3, 1]]. Find |A|.
- |A| = (5)(1) - (-2)(3)
- |A| = 5 + 6
- |A| = 11
- 11 != 0 so A is non-singular
In simple words
ad - bc, not ab - cd. The minus sits between the two diagonal products.
Key points
- ad - bc
- 0 => singular
- invertible iff |A|!=0
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.DETERMINANT OF A 2x2 MATRIX
Sticky Notes · Quick Revision
Core Idea
- For A = [[a, b], [c, d]], |A| = ad - bc. It is a single number, not a matrix. |A|=0 means singular (no inverse). |A| != 0 means non-singular (inverse exists). A diagonal 2x2 has |A| = product of the diagonal entries.
COMPUTE
- [[3,1],[2,4]]: 3*4 - 1*2 = 10. [[1,2],[3,4]]: 4-6=-2. [[2,4],[1,2]]: 4-4=0, singular.
WHAT |A| TELLS YOU
- Non-zero: inverse exists (next page). Zero: do not write A-1. Determinant is not the same as adding the diagonal (that is the trace, not in this chapter's main list).
Worked Method
- Worked: A = [[5, -2], [3, 1]]. Find |A|.
- |A| = (5)(1) - (-2)(3)
- |A| = 5 + 6
- |A| = 11
- 11 != 0 so A is non-singular
Common Misconception
- ad - bc, not ab - cd. The minus sits between the two diagonal products.
FBISE Exam Check
- ad - bc, not ab - cd. The minus sits between the two diagonal products.
60-second revision
- For A = [[a, b], [c, d]], |A| = ad - bc.
- COMPUTE: [[3,1],[2,4]]: 3*4 - 1*2 = 10.
- WHAT |A| TELLS YOU: Non-zero: inverse exists (next page).
- Worked-method reminder: 11 != 0 so A is non-singular
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