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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.
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DETERMINANT OF A 2x2 MATRIX
What is it?
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.
In simple words
- 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
Important terms
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).
What you should learn
- 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.
Examples
- 3*4-1*2=10
- 1*4-2*3=-2
- 2*2-4*1=0
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).
Recommended for Class 10
Full school-level explanation
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.
Main diagonal minus the other:
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.
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.
Remember
- ad - bc
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).
Remember
- invertible iff |A|!=0
Examples
- |A|=10 invertible
- |A|=0 no inverse
- Not a+d
Remember
- ad - bc
- 0 => singular
- |A| = 11
- ad - bc
- ad - bc
- invertible iff |A|!=0
- ad - bc
- 0 => singular
How the ideas connect
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
Deeper understanding
Go beyond memorising DETERMINANT OF A 2x2 MATRIX. Connect the definition to reasons, relationships, exceptions, and exam wording.
How the ideas connect
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 mistakes
Exam focus
Core Idea
COMPUTE
WHAT |A| TELLS YOU
Worked Method
Exam focus
Quick revision
- ad - bc
- 0 => singular
- |A| = 11
- 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 (
- 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 addi
ad - bc, not ab - cd. The minus sits between the two diagonal products.
For long answers on DETERMINANT OF A 2x2 MATRIX: define clearly → explain the main parts → give one example → state one exam caution.
Exam Focus
Exam Focus
Important points
- ad - bc
- 0 => singular
- |A| = 11
- 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 (
Important formulas
- ad - bc
- 0 => singular
- |A| = 11
Common exam traps
- ad - bc, not ab - cd. The minus sits between the two diagonal products.
- Not a+d
Long question structure
Definition → key parts → example → exam caution
Common Mistakes
Common Mistakes
- Mistake
ad - bc, not ab - cd. The minus sits between the two diagonal products.
- Mistake
Not a+d
- Mistake
Do not treat COMPUTE and WHAT |A| TELLS YOU as the same step; each has a different purpose.
- Mistake
ad - bc, not ab - cd. The minus sits between the two diagonal products.
Revision Zone
60-Second Revision
Quick cues from this topic — use them for last-minute revision.
ad - bc
0 => singular
|A| = 11
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 (
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
0 => singular
- 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
- Formula
- 0 => singular
- Example
- 3*4-1*2=10
- Critical warning
- ad - bc, not ab - cd. The minus sits between the two diagonal products.
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Exam Practice
Define
Q1. Define DETERMINANT OF A 2x2 MATRIX.
Explain
Q2. Explain: COMPUTE.
Example
Q3. Give one clear example of DETERMINANT OF A 2x2 MATRIX and say why it fits.
Mistake
Q4. State one common mistake students make about DETERMINANT OF A 2x2 MATRIX.
Exam note
Q5. Write a short exam note on DETERMINANT OF A 2x2 MATRIX: definition, two points, one example.
Sources & References
School-standard facts for Matric and Intermediate. Original explanations — not copied from textbooks.