Fundamentals of Engineering Analysis EGR 1302 - Determinants Approximate Running Time - 22 minutes

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© 2005 Baylor University Slide 1 Fundamentals of Engineering Analysis EGR 1302 - Determinants Approximate Running Time - 22 minutes Distance Learning / Online Instructional Presentation Presented by Department of Mechanical Engineering Baylor University Procedures: 1. Select “Slide Show” with the menu: Slide Show| View Show (F5 key), and hit “Enter” 2. You will hear “CHIMES” at the completion of the audio portion of each slide; hit the “Enter” key, or the “Page Down” key, or “Left Click” 3. You may exit the slide show at any time with the “Esc” key; and you may select and replay any slide, by navigating with the “Page Up/Down” keys, and then hitting “Shift+F5”.

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Fundamentals of Engineering Analysis EGR 1302 - Determinants Approximate Running Time - 22 minutes Distance Learning / Online Instructional Presentation Presented by Department of Mechanical Engineering Baylor University. Procedures: - PowerPoint PPT Presentation

Transcript of Fundamentals of Engineering Analysis EGR 1302 - Determinants Approximate Running Time - 22 minutes

Page 1: Fundamentals of Engineering Analysis EGR 1302 - Determinants Approximate Running Time - 22 minutes

© 2005 Baylor UniversitySlide 1

Fundamentals of Engineering AnalysisEGR 1302 - Determinants

Approximate Running Time - 22 minutesDistance Learning / Online Instructional Presentation

Presented byDepartment of Mechanical Engineering

Baylor University

Procedures:

1. Select “Slide Show” with the menu: Slide Show|View Show (F5 key), and hit “Enter”

2. You will hear “CHIMES” at the completion of the audio portion of each slide; hit the “Enter” key, or the “Page Down” key, or “Left Click”

3. You may exit the slide show at any time with the “Esc” key; and you may select and replay any slide, by navigating with the “Page Up/Down” keys, and then hitting “Shift+F5”.

Page 2: Fundamentals of Engineering Analysis EGR 1302 - Determinants Approximate Running Time - 22 minutes

© 2005 Baylor UniversitySlide 2

Determinants

333231

232221

131211

det

aaa

aaa

aaa

A“Eyeball” Method

122133113223312213122331133221332211det aaaaaaaaaaaaaaaaaaA

3 positive terms 3 negative terms

- A Property of a Square Matrix

Page 3: Fundamentals of Engineering Analysis EGR 1302 - Determinants Approximate Running Time - 22 minutes

© 2005 Baylor UniversitySlide 3

Determinant of a 3x3

Let’s factor out the elements of the first row of the matrix, i.e.

122133113223312213122331133221332211det aaaaaaaaaaaaaaaaaaA

)()()(det 223132211323313321123223332211 aaaaaaaaaaaaaaaA

333231

232221

131211

det

aaa

aaa

aaa

A

Page 4: Fundamentals of Engineering Analysis EGR 1302 - Determinants Approximate Running Time - 22 minutes

© 2005 Baylor UniversitySlide 4

)()()(det 223132211323313321123223332211 aaaaaaaaaaaaaaaA

Determinant of a 3x3

333231

232221

131211

det

aaa

aaa

aaa

A

3231

222113

3331

232112

3332

232211det

aa

aaa

aa

aaa

aa

aaaA

We can identify this construct as the “Cofactor”

Page 5: Fundamentals of Engineering Analysis EGR 1302 - Determinants Approximate Running Time - 22 minutes

© 2005 Baylor UniversitySlide 5

The Cofactor Matrix of a 3x3

The cofactor of any element is “the determinant formed by striking out the Row & Column of that element

3332

232211 aa

aaaCofactor

2221

1211

2321

1311

2322

1312

3231

1211

3331

1311

3332

1312

3231

2221

3331

2321

3332

2322

aa

aa

aa

aa

aa

aaaa

aa

aa

aa

aa

aaaa

aa

aa

aa

aa

aa

AC f

333231

232221

131211

aaa

aaa

aaa

AEvery element in a square matrix has a cofactor

Page 6: Fundamentals of Engineering Analysis EGR 1302 - Determinants Approximate Running Time - 22 minutes

© 2005 Baylor UniversitySlide 6

2221

1211

2321

1311

2322

1312

3231

1211

3331

1311

3332

1312

3231

2221

3331

2321

3332

2322

aa

aa

aa

aa

aa

aaaa

aa

aa

aa

aa

aaaa

aa

aa

aa

aa

aa

AC f

333231

232221

131211

aaa

aaa

aaa

A

)()()(det 223132211323313321123223332211 aaaaaaaaaaaaaaaA

The Cofactor Matrix of a 3x3

Sign of the Cofactor:

12aaij

)( evenji

)( oddjinm )1(

)(,21 odd

Caution: Do not forget the signs of the cofactors

Page 7: Fundamentals of Engineering Analysis EGR 1302 - Determinants Approximate Running Time - 22 minutes

© 2005 Baylor UniversitySlide 7

3231

222113

3331

232112

3332

232211det

aa

aaa

aa

aaa

aa

aaaA

Determinant by Row Expansion

333231

232221

131211

aaa

aaa

aaa

A

)()()(det 223132211323313321123223332211 aaaaaaaaaaaaaaaA

Row Expansion:

960310

13*2

20

13*0

21

11*1

210

113

201

det

A

using the first row:

Page 8: Fundamentals of Engineering Analysis EGR 1302 - Determinants Approximate Running Time - 22 minutes

© 2005 Baylor UniversitySlide 8

Using the TI-89 to find Determinants

We had previously entered a matrixand assigned it to the variable “a”

The calculator has the built-in function “det()“Which calculates the determinant of a square matrix.

Page 9: Fundamentals of Engineering Analysis EGR 1302 - Determinants Approximate Running Time - 22 minutes

© 2005 Baylor UniversitySlide 9

4)46(*231

42*2

310

420

202

det A

Determinant by Row or Column Expansion

Select Any Row or Column to do the Expansion

Pick Column #1 to simplify the calculation due to the zero terms.

Page 10: Fundamentals of Engineering Analysis EGR 1302 - Determinants Approximate Running Time - 22 minutes

© 2005 Baylor UniversitySlide 10

210

113

201

A

13

01

13

21

11

2010

01

20

21

21

2010

13

20

13

21

11

AC f

Finding the Cofactor Matrix of A

172

122

363

AC f

Calculators and Computers obviously make this process easier.

Page 11: Fundamentals of Engineering Analysis EGR 1302 - Determinants Approximate Running Time - 22 minutes

© 2005 Baylor UniversitySlide 11

Rules for 2x2 Inverse and the Cofactor Matrix

2221

1211

aa

aaA

1112

2122

aa

aaAC f

1. Swap Main Diagonal

2. Change Signs on a12, a21

Aaa

aaA

det

1*

1121

12221

3. Divide by detA

Similar, but not quite

Page 12: Fundamentals of Engineering Analysis EGR 1302 - Determinants Approximate Running Time - 22 minutes

© 2005 Baylor UniversitySlide 12

Properties of Determinants

1. Determinant of the Transpose Matrix

det A = det AT

210

113

201

A

212

110

031TA

10

13*2

21

11*1det

A

11

03*2

21

11*1det

TA

Page 13: Fundamentals of Engineering Analysis EGR 1302 - Determinants Approximate Running Time - 22 minutes

© 2005 Baylor UniversitySlide 13

Properties of Determinants

2. Multiply a single Row (Column) by a Scalar - k

333231

232221

131211

aaa

aaa

aaa

A

333231

232221

131211 ***

aaa

aaa

akakak

B

det B = k*det A

210

113

201

A

210

113

603

B

9det A

3kfor

27det B

det B = 3*det A

Page 14: Fundamentals of Engineering Analysis EGR 1302 - Determinants Approximate Running Time - 22 minutes

© 2005 Baylor UniversitySlide 14

Properties of Determinants

3. If two Rows (Columns) are swapped, the sign changes

det B = -det A

333231

131211

232221

aaa

aaa

aaa

Bswap

333231

232221

131211

aaa

aaa

aaa

A

Recall:

10

13*2

21

11*1det

A

10

13*2

21

11*1det

TA

3337

34,33

73

43,33

43

73

DCA

4. Expansion by any Rows (Columns) equals the same Determinant

Page 15: Fundamentals of Engineering Analysis EGR 1302 - Determinants Approximate Running Time - 22 minutes

© 2005 Baylor UniversitySlide 15

Properties of Determinants

5. If two Rows (Columns) are equal, or the same ratio,i.e., Row1 = k*Row2

det A = 0

200

163

221

B

det B = 0

Col2 = 2*Col1

210

113

113

A

det A = 0

Row2 = Row1

The matrix A is “singular”

Recall Rule #3 to find A-1,divide by detA

But if detA=0,a unique solution does not exist

Page 16: Fundamentals of Engineering Analysis EGR 1302 - Determinants Approximate Running Time - 22 minutes

© 2005 Baylor UniversitySlide 16

Properties of Determinants

6. If a new matrix B is constructed from Aby adding K*rowj to another rowi …

det B = det A

jii rowkrowrow *'

67

32A 33det A

011

32D 33det D

12'

2 *2 rrr )]3*2(6[)],2*2(7['2 r

Construct D by creating a new Row 2

These are called Row (Column) Operations

Page 17: Fundamentals of Engineering Analysis EGR 1302 - Determinants Approximate Running Time - 22 minutes

© 2005 Baylor UniversitySlide 17

241

312

521

det

A

Finding the Determinant: Two Methods

2 + (-40) + (-6) – (-5) -12 –(-8) = -43

“Eyeball” Method

Row Expansion41

12*)5(

21

32*2

24

31*1

1*(2-12) -2(-4+3) -5(8-1)

-10 + 2 -35 = -43

Page 18: Fundamentals of Engineering Analysis EGR 1302 - Determinants Approximate Running Time - 22 minutes

© 2005 Baylor UniversitySlide 18

This concludes the Lecture