Fundamentals of Engineering Analysis EGR 1302 - Determinants Approximate Running Time - 22 minutes
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Transcript of 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
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© 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
© 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
© 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”
© 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
© 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
© 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:
© 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.
© 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.
© 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.
© 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
© 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
© 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
© 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
© 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
© 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
© 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
© 2005 Baylor UniversitySlide 18
This concludes the Lecture