1. 3-1 3.1 The Determinant of a Matrix Note: The determinant of a matrix can be positive, zero, or negative. Chapter 3 Determinants

2. 3-2 Notes: Sign pattern for cofactors

3. 3-3

4. 3-4

5. 3-5 The determinant of a matrix of order 3: Add these three products. Subtract these three products.

6. 3-6 Upper triangular matrix: Lower triangular matrix: Diagonal matrix: All the entries below the main diagonal are zeros. All the entries above the main diagonal are zeros. All the entries above and below the main diagonal are zeros. Ex: upper triangularlower triangulardiagonal

7. 3-7 A row-echelon form of a square matrix is always upper triangular.

8. 3-8 3.2 Evaluation of a Determinant Using Elementary Operations

9. 3-9

10. 3-10 Determinants and Elementary Column Operations: The elementary row operations can be replaced by the column operations and two matrices are called column-equivalent if one can be obtained form the other by elementary column operations.

11. 3-11

12. 3-12 3.3 Properties of Determinants Notes: (1) (2)

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15. 3-15

16. 3-16

17. 3-17

18. 3-18 3.4 Applications of Determinants Matrix of cofactors of A: Adjoint matrix of A:

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20. 3-20

21. 3-21

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24. 3-24