CS223 · Annotated SlidesChapter 3 — Cramer's Rule, the Inverse Formula, Area and Volume
Chapter 3 · Annotated slides

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.

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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)
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  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.

Related real-exam questions

Major 2 (251) · this topic is examined
🎯 How this can come up · 1 question
Written

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

Slide 2: Slide 2
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    Slide 3: Slide 3
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      Slide 4: Slide 4
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        Slide 5: The area after transformation is equal to the area before transformation multiplied by the det of A
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        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|.
        🎯 How this can come up · 1 question
        MCQ

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

        Slide 6: Practice problem 1
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          Slide 7: Practice problem 2
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            Practice problem 3

            Show that if A is 2 x 2, then Theorem 8 gives the same formula for A as the one given in Section 2.2

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              Slide 9: Practice problem 4
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                Practice problem 5

                Find the area of the parallelogram whose vertices are listed:

                (0,0) , (5,2) , (6,4) , (11,6) (1, 0),(2, 5),(3, – 4), and (4, 1)

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