Download presentation

Presentation is loading. Please wait.

Published byHunter Rodriguez Modified over 4 years ago

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?

Similar presentations

OK

MATRICES Operations with Matrices Properties of Matrix Operations

MATRICES Operations with Matrices Properties of Matrix Operations

© 2018 SlidePlayer.com Inc.

All rights reserved.

To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy, including cookie policy.

Ads by Google