1 / 13

Chapter Content

Chapter Content. Determinants by Cofactor Expansion Evaluating Determinants by Row Reduction Properties of the Determinant Function A Combinatorial Approach to Determinants. Theorems. Theorem 2.2.1 Let A be a square matrix

josiah-dillard
Download Presentation

Chapter Content

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Chapter Content • Determinants by Cofactor Expansion • Evaluating Determinants by Row Reduction • Properties of the Determinant Function • A Combinatorial Approach to Determinants

  2. Theorems • Theorem 2.2.1 • Let A be a square matrix • If A has a row of zeros or a column of zeros, then det(A) = 0. • Theorem 2.2.2 • Let A be a square matrix • det(A) = det(AT)

  3. Theorem 2.2.3 (Elementary Row Operations) • Let A be an nn matrix • If B is the matrix that results when a single row or single column of A is multiplied by a scalar k, than det(B) = k det(A) • If B is the matrix that results when two rows or two columns of A are interchanged, then det(B) = - det(A) • If B is the matrix that results when a multiple of one row of A is added to another row or when a multiple column is added to another column, then det(B) = det(A) • Example 1

  4. 2-2 Example of Theorem 2.2.3

  5. Theorem 2.2.4 (Elementary Matrices) • Let E be an nn elementary matrix • If E results from multiplying a row of In by k, then det(E) = k • If E results from interchanging two rows of In, then det(E) = -1 • If E results from adding a multiple of one row of In to another, then det(E) = 1 • Example 2

  6. Theorem 2.2.5 (Matrices with Proportional Rows or Columns) • If A is a square matrix with two proportional rows or two proportional column, then det(A) = 0 • Example 3

  7. 2-2 Example 4 (Using Row Reduction to Evaluate a Determinant) • Evaluate det(A) where • Solution: The first and second rows of A are interchanged. A common factor of 3 from the first row was taken through the determinant sign

  8. 2-2 Example 4 (continue) -2 times the first row was added to the third row. -10 times the second row was added to the third row A common factor of -55 from the last row was taken through the determinant sign.

  9. 2-2 Example 5 • Using column operation to evaluate a determinant • Compute the determinant of

  10. 2-2 Example 6 • Row operations and cofactor expansion • Compute the determinant of

  11. Exercise Set 2.2 Question 12

  12. Exercise Set 2.2 Question 13

  13. Exercise Set 2.2 Question 20

More Related