Fundamentals of Engineering Analysis EGR 1302 - Adjoint Matrix and Inverse Solutions
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© 2005 Baylor University Slide 1 Fundamentals of Engineering Analysis EGR 1302 - Adjoint Matrix and Inverse Solutions Approximate Running Time - 7 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 - Adjoint Matrix and Inverse Solutions Approximate Running Time - 7 minutes Distance Learning / Online Instructional Presentation Presented by Department of Mechanical Engineering Baylor University. Procedures: - PowerPoint PPT Presentation
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© 2005 Baylor UniversitySlide 1
Fundamentals of Engineering AnalysisEGR 1302 - Adjoint Matrix and Inverse Solutions
Approximate Running Time - 7 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”.
© 2005 Baylor UniversitySlide 2
The Adjoint Matrix and the Inverse Matrix
Recall the Rules for the Inverse of a 2x2:1. Swap Main Diagonal2. Change sign of a12, a21 3. Divide by determinant
2221
1211
aa
aaA
Aaa
aaA
det
1*
1121
12221
1112
2122
aa
aaAC f
If the Cofactor Matrix is “transposed”,we get the same matrix as the Inverse
1121
1222
aa
aaAC Tf
1121
1222
aa
aaadjAAnd we define the “Adjoint” as the
“Transposed Matrix of Cofactors”.
A
adjAA
det1
And we see that the Inverse is defined as
© 2005 Baylor UniversitySlide 3
Calculating the Adjoint Matrix and A-1
22
02
22
12
22
1040
02
10
12
14
1040
22
10
22
14
22
AC f
422
824
8210
AA
det
488
222
2410
1
-12detA
140
222
102
A Problem 7.13 in the Text
adjA =
© 2005 Baylor UniversitySlide 4
Complexity of Large Matrices
20311
12712
31040
22123
11201
SConsider the 5x5 matrix, S
To find the Adjoint of S (in order to find the inverse), would require
• Finding the determinants of 25 4x4s, which means• Finding the determinants of 25*16 = 400 3x3s, which means• Finding the determinants of 400*9 = 3600 2x2s. (Wow!)
Which is why we use computers(and explains why so many problems could not be
solved before the advent of computers).
© 2005 Baylor UniversitySlide 5
Class Exercise: Find the Adjoint of A
987
654
321
A
Work this out yourself before going to the solution on the next slide
?adjA
© 2005 Baylor UniversitySlide 6
Class Exercise: Solution
363
6126
363
adjA
Notice that: detA = 0, therefore matrix A is singular.
However, even though the Determinant is zero, the Adjoint still exists.
This means that the Inverse does not exist.A
adjAA
det1
© 2005 Baylor UniversitySlide 7
This concludes the Lecture
