CS223 · Annotated SlidesChapter 3 — Properties of Determinants
Chapter 3 · Annotated slides

Properties of Determinants

How row operations change the determinant, computing det by row reduction, det A ≠ 0 ⇔ A invertible, det Aᵀ = det A and det(AB) = det A · det B.

Slide text Study notes Highlight: key words Tips, tricks & traps Seen again elsewhere How it can come up
Slide 1: Determinant of invertible and noninvertible matrices
Exam weight High
  1. Row replacement: det unchanged. Row interchange: det changes sign. Scaling a row by k: det multiplied by k.
Memory trickReplace → same; Swap → minus; Scale by k → times k.
🎯 How this can come up · 1 question
MCQ

If B is A with two rows swapped and det A = 5, det B =

Determinant of invertible and noninvertible matrices

2
Exam weight Low
    Slide 3: Slide 3
    Exam weight Low
      Slide 4: Slide 4
      Exam weight Low

        5
        Exam weight Low

          6
          Exam weight Low

            Slide 7
            7
            Exam weight Low
              Slide 8: Slide 8
              Exam weight High
              1. A is invertible ⇔ det A ≠ 0. det Aᵀ = det A. det(AB) = det A · det B. det(kA) = kⁿ det A for n×n A. det A⁻¹ = 1/det A.
              Exam trapdet(A + B) ≠ det A + det B, and det(kA) = kⁿ·det A, not k·det A.
              🎯 How this can come up · 1 question
              MCQ

              A is 3×3 with det A = 2. det(2A) =

              Slide 9: Practice problem 1
              Exam weight Low

                Practice problem 2

                Use Cramer ’ s rule to compute the solutions of the following systems:

                a) b)

                Practice problem 2
                10
                Exam weight Low