1. BAI CM20144 Applications I: Mathematics for Applications Mark Wood cspmaw@cs.bath.ac.uk http://www.cs.bath.ac.uk/~cspmaw

2. BAI Determinants Evaluation Methods Properties Examples Test 5 Todays Tutorial

3. BAI Evaluating Determinants 1

4. BAI Diagonals Method Only works for 2 x 2 and 3 x 3 Multiply forward diagonal elements and add Multiply backward diagonal elements and subtract Evaluating Determinants 1

5. BAI Diagonals Method Only works for 2 x 2 and 3 x 3 Multiply forward diagonal elements and add Multiply backward diagonal elements and subtract Cofactor Method Pick the row or column with the most zeros Calculate the cofactor for each element and sum Cofactor = sign x minor Signs alternate Minor = determinant of remaining matrix… Evaluating Determinants 1

6. BAI Diagonals Method Only works for 2 x 2 and 3 x 3 Multiply forward diagonal elements and add Multiply backward diagonal elements and subtract Cofactor Method Pick the row or column with the most zeros Calculate the cofactor for each element and sum Cofactor = sign x minor Signs alternate Minor = determinant of remaining matrix… Recursive Evaluating Determinants 1

7. BAI 3 1 4 -7 -2 1 9 1 -1 Example: Diagonals

8. BAI 3 1 4 -7 -2 1 9 1 -1 Example: Cofactors

9. BAI Properties of Determinants

10. BAI Singular Matrices Determinant = 0 (otherwise nonsingular) Row or column of zeros singular Two rows proportional singular Properties of Determinants

11. BAI Singular Matrices Determinant = 0 (otherwise nonsingular) Row or column of zeros singular Two rows proprtional singular Invertible nonsingular Properties of Determinants

12. BAI Singular Matrices Determinant = 0 (otherwise nonsingular) Row or column of zeros singular Two rows proprtional singular Invertible nonsingular Other properties Scalar multiple: |cA| = c n |A|(n = matrix dim) Product: |AB| = |A||B| Transpose: |A t | = |A| Inverse: |A -1 | = 1/|A|(if A -1 exists) Properties of Determinants

13. BAI A and B are 3 x 3 matrices |A| = -3, |B| = 2 Calculate: |AB| |AA t | |A t B| |3A 2 B| |2AB -1 | |(A 2 B -1 ) t | Example: Properties of Determinants

14. BAI Evaluating Determinants 2

15. BAI Row Operations and Determinants 1) Multiply by c c|A| 2) Swap two rows -|A| 3) Add multiple of one row to another |A| Evaluating Determinants 2

16. BAI Row Operations and Determinants 1) Multiply by c c|A| 2) Swap two rows -|A| 3) Add multiple of one row to another |A| Get zero columns / rows and use cofactors Evaluating Determinants 2

17. BAI Row Operations and Determinants 1) Multiply by c c|A| 2) Swap two rows -|A| 3) Add multiple of one row to another |A| Get zero columns / rows and use cofactors Numerical Method Use row ops to get matrix into upper triangular form Only need 2) and 3) Keep track of op 2) Determinant is product of diagonal elements Zero on diagonal & zeros below singular Evaluating Determinants 2

18. BAI 1 0 –2 1 2 1 0 2 -1 1 –2 1 3 1 –1 0 Example: Numerical Evaluation

19. BAI 1 -1 0 2 -1 1 0 0 2 -2 0 1 3 1 5 -1 Example: Numerical Evaluation

20. BAI Other Stuff?

21. BAI A -1 = adj(A) / |A| Adjoint is transpose of matrix of cofactors Other Stuff?

22. BAI A -1 = adj(A) / |A| Adjoint is transpose of matrix of cofactors System of Equations AX = B Unique solution A nonsingular Otherwise, could be many or no solutions Cramers Rule: x i = |A i | / |A| Other Stuff?