CS223 · Chapter 3 · Annotated SlidesCramer's Rule, the Inverse Formula, Area and Volume
Original
Overview

Cramer's Rule, the Inverse Formula, Area and Volume

Cramer's rule xᵢ = det Aᵢ(b) / det A, the adjugate formula A⁻¹ = adj A / det A, and determinants as areas and volumes, including how a linear transformation scales area.

How to read this page

Every original slide is shown unchanged. Yellow marks on a slide point to a numbered note: hover or tap a mark to jump to its note, and hover a note to see its marks. Questions tagged Past exam appeared on a real paper; Prediction questions are likely angles, not confirmed. Pick an answer to each question: a wrong pick tells you why it is wrong, so you can try again.

All 10 slides
  1. 1 Steps: 1. Calculate cofactor at position ( i , j ) of the original matrix A . 2. Assign result to position ( j, i ) (opposite order of i and j) in the adjugate matrix (adj A)
  2. 2 Slide 2
  3. 3 Slide 3
  4. 4 Slide 4
  5. 5 The area after transformation is equal to the area before transformation multiplied by the det of A
  6. 6 Practice problem 1
  7. 7 Practice problem 2
  8. 8 Practice problem 3
  9. 9 Practice problem 4
  10. 10 Practice problem 5
Slide 1 / 10

Steps: 1. Calculate cofactor at position ( i , j ) of the original matrix A . 2. Assign result to position ( j, i ) (opposite order of i and j) in the adjugate matrix (adj A)

Slide 1: Steps: 1. Calculate cofactor at position ( i , j ) of the original matrix A . 2. Assign result to position ( j, i ) (opposite order of i and j) in the adjugate matrix (adj A)

Notes

  1. Cramer's rule: xᵢ = det Aᵢ(b) / det A, where Aᵢ(b) replaces column i with b. A⁻¹ = (1/det A)·adj A, adj A = transpose of the cofactor matrix.

Exam angle

Past exam · Major 2 (251)

Related question: this topic is examined

PredictionQ1

Use Cramer's rule: 3x₁ − 2x₂ = 6, −5x₁ + 4x₂ = 8.

Compare with the model answer

det A = 12 − 10 = 2. det A₁ = 6·4 − (−2)·8 = 40 → x₁ = 20. det A₂ = 3·8 − 6·(−5) = 54 → x₂ = 27.

Slide 2 / 10

Slide 2

Slide 2: Slide 2
Slide 3 / 10

Slide 3

Slide 3: Slide 3
Slide 4 / 10

Slide 4

Slide 4: Slide 4
Slide 5 / 10

The area after transformation is equal to the area before transformation multiplied by the det of A

Slide 5: The area after transformation is equal to the area before transformation multiplied by the det of A

Notes

  1. |det A| = area of the parallelogram (2×2) or volume of the parallelepiped (3×3) spanned by the columns. A transformation multiplies areas by |det A|.

Exam angle

PredictionQ1

T(x) = Ax with det A = −3 maps a region of area 4 to area:

Slide 6 / 10

Practice problem 1

Slide 6: Practice problem 1
Slide 7 / 10

Practice problem 2

Slide 7: Practice problem 2
Slide 8 / 10

Practice problem 3

Slide 8: Practice problem 3
Slide 9 / 10

Practice problem 4

Slide 9: Practice problem 4
Slide 10 / 10

Practice problem 5

Slide 10: Practice problem 5