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(AB)⁻¹ = B⁻¹A⁻¹ – Multi-Matrix Inverse Exam Technique | Matrices | AS Level Further Mathematics (9231) | Educate A Change, Hunain Zia (AYLOTI) Preparation Notes

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Outline

  • Why (AB)⁻¹ = B⁻¹A⁻¹ Is a Core Identity in AS Level Further Mathematics (9231)
  • Understanding the Order Reversal in Matrix Inverses
  • Proof of (AB)⁻¹ = B⁻¹A⁻¹ Using Identity Matrix Logic
  • Conditions Required for Multi-Matrix Inverses
  • Step-by-Step Exam Technique for (AB)⁻¹ Questions
  • Extending to Three or More Matrices
  • Determinant Perspective on Product Inverses
  • Using the Identity in Matrix Equation Problems
  • Examiner Report Based Errors in Multi-Matrix Inverse Questions
  • Structured Full-Marks Layout for Product Inverse Problems
  • Application in Transformation and Algebraic Matrix Questions
  • Time Strategy for Multi-Matrix Inverse Questions
  • Final Accuracy Checklist for Product of Matrices Inverse

Why (AB)⁻¹ = B⁻¹A⁻¹ Is a Core Identity in AS Level Further Mathematics (9231)

  • In AS Level Further Mathematics (9231), the identity:
    • (AB)⁻¹ = B⁻¹A⁻¹
      is heavily tested
  • It appears in:
    • Proof-based questions
    • Simplification tasks
    • Matrix equation rearrangements
  • Students often:
    • Memorise formula
    • Forget why the order reverses
  • Examiners test:
    • Conceptual clarity
    • Not blind substitution

Understanding order reversal is essential.

Understanding the Order Reversal in Matrix Inverses

Matrix multiplication is:

  • Not commutative
  • AB ≠ BA

When taking inverse:

  • We need matrix that satisfies:
    • (AB)(AB)⁻¹ = I

If we multiply:

  • (AB)(B⁻¹A⁻¹)

Then:

  • A(BB⁻¹)A⁻¹
  • AIA⁻¹
  • AA⁻¹
  • I

This proves:

  • (AB)⁻¹ = B⁻¹A⁻¹

Order reversal ensures identity formation.

Proof of (AB)⁻¹ = B⁻¹A⁻¹ Using Identity Matrix Logic

Proof structure:

  • Start with:
    • (AB)(B⁻¹A⁻¹)
  • Rearrange brackets carefully
  • Use:
    • BB⁻¹ = I
    • AA⁻¹ = I
  • Show result equals:
    • Identity matrix

Examiners expect:

  • Clear logical steps
  • No skipped reasoning

Conditions Required for Multi-Matrix Inverses

For (AB)⁻¹ to exist:

  • A must be invertible
  • B must be invertible
  • Determinants must satisfy:
    • |A| ≠ 0
    • |B| ≠ 0

If either matrix is singular:

  • Product inverse does not exist

Students lose marks by:

  • Applying identity without checking invertibility

Step-by-Step Exam Technique for (AB)⁻¹ Questions

When question asks:

  • Find (AB)⁻¹

Method:

  • Step 1: Find A⁻¹
  • Step 2: Find B⁻¹
  • Step 3: Reverse order
  • Step 4: Multiply B⁻¹A⁻¹

Never:

  • Multiply AB first unless required

Reversing order is mandatory.

Written and Compiled By Sir Hunain Zia (AYLOTI), World Record Holder With 154 Total A Grades, 11 World Records and 7 Distinctions, Educate A Change

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Extending to Three or More Matrices

For three matrices:

  • (ABC)⁻¹ = C⁻¹B⁻¹A⁻¹

Pattern:

  • Reverse entire order

Students often:

  • Reverse partially
  • Write A⁻¹B⁻¹C⁻¹

Examiners penalise:

  • Incorrect order immediately

Complete reversal is required.

Determinant Perspective on Product Inverses

Determinant property:

  • |AB| = |A||B|

If:

  • |AB| ≠ 0
  • Then both |A| ≠ 0 and |B| ≠ 0

This confirms invertibility condition

Students must understand:

  • Determinant logic supports inverse identity

Using the Identity in Matrix Equation Problems

Typical form:

  • ABX = C

Rearrangement:

  • X = B⁻¹A⁻¹C

Students often:

  • Multiply in wrong order
  • Forget reversal rule

Careful algebra prevents loss of marks.

Examiner Report Based Errors in Multi-Matrix Inverse Questions

Common examiner comments:

  • Students forgetting to reverse order
  • Incorrect inverse calculation
  • Arithmetic mistakes in multiplication
  • Skipping determinant checks

Most errors are structural.

Structured Full-Marks Layout for Product Inverse Problems

High-scoring structure:

  • State invertibility condition
  • Find A⁻¹ clearly
  • Find B⁻¹ clearly
  • Reverse order explicitly
  • Multiply carefully
  • Present final matrix neatly

Clarity protects method marks.

Written and Compiled By Sir Hunain Zia (AYLOTI), World Record Holder With 154 Total A Grades, 11 World Records and 7 Distinctions, Educate A Change

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Application in Transformation and Algebraic Matrix Questions

Matrices represent:

  • Linear transformations
  • Coordinate mapping

When transformation matrices multiply:

  • Order represents sequence

Inverse of combined transformation:

  • Reverse sequence

Conceptual understanding supports algebra accuracy.

Time Strategy for Multi-Matrix Inverse Questions

  • These questions are:
    • Method-heavy
  • Spend time:
    • On correct inversion
  • Avoid:
    • Rushing multiplication
  • Recheck:
    • Order reversal

Most lost marks are avoidable.

Final Accuracy Checklist for Product of Matrices Inverse

  • Determinant non-zero checked
  • Individual inverses correct
  • Order fully reversed
  • Multiplication structured clearly
  • Final matrix simplified properly

Product inverse mastery in 9231 depends entirely on respecting order and structure.

Written and Compiled By Sir Hunain Zia (AYLOTI), World Record Holder With 154 Total A Grades, 11 World Records and 7 Distinctions, Educate A Change

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Hunain Zia

Hunain Zia has previously achieved considerable success in the high-school educational stream, gaining a massive 154 Total A Grades and 7 Major Distinctions, a process in which he broke/ set 11 different world records, including the most A grades ever achieved, most subjects ever appeared in, most distinctions, gaining distinctions across two separate boards (Pearson Edexcel and CAIE) within the same year/ ever, and even 19 hours of constant examination. All records stand intact today. Hunain pursued his Honors Accounting Degree from the prestigious Bentley University, and then completed an LLB (in Honors) from University of London. Currently, he manages multiple social and commercial projects, has founded digital streams and works tirelessly in the education sector.

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