Introducing the Determinant
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INTRODUCING THE DETERMINANT
C. Ray RosentraterWestmont College
2013 Joint Mathematics Meetings
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PREMISE
When students are introduced to a new concept via a problem they understand:
1. They can be engaged in exploratory/active learning exercises.
2. They understand the new concept better.
3. They are more willing to engage in theoretical analysis of the concept.
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WHAT ARE THE COMMON APPROACHES?
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PRESENTATION OF THE DETERMINANT: TEXT 1
Motivation: Want to study a function with a matrix variable.
Development Thread:1. Permutations2. Elementary Products (Definition)3. Evaluation by Row Reduction (No
justification)4. Properties5. Cofactor Expansion (No justifiction)6. Application: Crammer’s Rule
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PRESENTATION OF THE DETERMINANT: TEXT 2
Motivation: Another important number associated with a square matrix.
Development Thread:1. Permutations2. Definition3. Properties (Row ops & Evaluation via
triangular matrices)4. Computation via Cofactors (3x3
justified)5. Applications: Crammer’s rule
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PRESENTATION OF THE DETERMINANT: TEXT 3
Motivation: List of uses (Singularity test, Volume, Sensitivity analysis)
Development Thread:1. Properties
1. Identity matrix, row exchange, linear in row one
2. Zero row, duplicate rows, triangular matrices, product rule, transpose (proved from first set)
2. Computation: Permutations and Cofactors
3. Applications: Cramer’s rule, Volume
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PRESENTATION OF THE DETERMINANT: TEXT 4
Motivation: Associate a real number to a matrix A in such a way that we can tell if A is singular.
Development Thread:1. 2x2, 3x3 singularity testing2. Cofactor Definition3. Properties (Row operations, Product)4. Applications: Crammer’s rule, Matrix
codes, Cross product
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PRESENTATION OF THE DETERMINANT: TEXT 5
Motivation: Singularity testingDevelopment Thread:1. 2x2, 3x3 singularity testing2. Cofactor Definition3. Properties: Row operations (not
justified), Products, Transposes4. Applications: Crammer’s rule, Volume,
Transformations
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SINGULARITY CHECKING (TEXT 5)
a11 a12
a21 a22
a11a22 a12a21
a11 a12 a13
a21 a22 a23
a31 a32 a33
a11a22a33 a12a23a31 a13a21a32 a11a23a32 a12a21a33 a13a22a31
a11 deta22 a23
a32 a33
a12 deta21 a23
a31 a33
a13 deta21 a22
a31 a32
Not amenable to active learningE. G. O.
~a11 a12
0 a22 a12a21a11
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PROPOSED PRESENTATION OF THE DETERMINANT
Motivation: Signed Area/Volume/Hyper-volume of the parallelogram (etc.) spanned by the rows
Development Thread:1. Simple Cases2. Row operations3. Semi-formal definition & computational
method4. Transition to Cofactor (Permutation)
Definition5. Properties
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SIMPLE CASES
a 0
0 b
a,b 0
a 0,b 0
a,b 0
deta 0
0 b ab
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SIMPLE CASES
x2
x3
x1
v1
v2
v3x2
x1
x3
v1 v2
v3
a 0 0
0 b 0
0 0 ca,b,c 0
a,c 0,b 0
det
a 0 0
0 b 0
0 0 c
abc
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PROPOSED PRESENTATION OF THE DETERMINANT
Motivation: Signed Area/Volume/Hyper-volume spanned by the rows
Development Thread:1. Simple Cases2. Row operations3. Semi-formal definition4. Transition to Cofactor (Permutation)
Definition5. Properties
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ROW OPERATIONS: ROW SCALING
v1
v1
v2
v2
sv1
sv1
v2v3
sv1v1
v2
v1
v2
v1
detB sdetA
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ROW OPERATIONS: ROW SWAP
v2
v1v2
v1
v2v3
v1
v2
v3
v1
detB detA
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ROW OPERATIONS: ROW REPLACEMENT
v2
v1
v2v3
v1
v2 + sv1 v3
v1
v2
detB detA
v2
v1
v1 + sv2v2 + sv1
v2
v1
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PROPOSED PRESENTATION OF THE DETERMINANT
Motivation: Signed Area/Volume/Hyper-volume spanned by the rows
Development Thread:1. Simple Cases2. Row operations3. Semi-formal definition & computation4. Transition to Cofactor (Permutation)
Definition5. Properties
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FIRST DEFINITION
Determinant = signed “volume” of the parallelogram spanned by the rows
To Compute: Use row replacements to put in triangular form, multiply the diagonal entries
det
1 2 3
2 3 5
3 4 6
1 2 3
2 3 5
3 4 6
~
1 2 3
0 1 1
0 2 3
~
1 2 3
0 1 1
0 0 1
~
1 2 0
0 1 0
0 0 1
~
1 0 0
0 1 0
0 0 1
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PROPOSED PRESENTATION OF THE DETERMINANT
Motivation: Signed Area/Volume/Hyper-volume spanned by the rows
Development Thread:1. Simple Cases2. Row operations3. Semi-formal definition4. Transition to Cofactor (Permutation)
Definition5. Properties
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TRANSITION TO COFACTOR DEFINITION
A I
2 1 3
1 4 5
3 2
2 1 3
1 4 5
3 2 0
1 0 0
0 1 0
0 0 1
Why have another method? A motivating example
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TRANSITION TO COFACTOR DEFINITION
2 1 3
1 4 5
3 2
~
2 1 3
0 4 12 5 3
2
0 2 32 9
2
Why have another method? A motivating example
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TRANSITION TO COFACTOR DEFINITION
~
2 1 3
0 4 12 5 3
2
0 2 32 9
2
~
2 1 3
0 4 12 5 3
2
0 0 92
5 32 2 3
2
4 12
detA I 2 4 12 9
2 5 3
2 2 32
4 12
3 6 2 12 35
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TRANSITION TO COFACTOR DEFINITION
• State the Cofactor definition• Verify the definitions agree
• Check simple (diagonal) case• Check row operation behavior
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ROW SCALING:
Verify scaling in row one from definition To scale row k
Swap row k with row one Scale row one Swap row k and row one
detB sdetA
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ROW REPLACEMENT:
det
a1,1 sak,1 a1,2 sak,1 a1,n sak,na2,1 a2,2 a2,n
an,1 an,2 an,n
det
a1,1 a1,2 a1,n
a2,1 a2,2 a2,n
an,1 an,2 an,n
sdet
ak,1 ak,2 ak,n
a2,1 a2,2 a2,n
an,1 an,2 an,n
det
a1,1 a1,2 a1,n
a2,1 a2,2 a2,n
an,1 an,2 an,n
• To add a multiple of row k to row j:• Swap rows one and j• Add the multiple of row k to row one• Swap rows one and j
detB detA
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SWAPPING FIRST TWO ROWSBA
Sign on a1,i a2,j A1,i; 2,j
a1,1 a1,2 a1, a1, a1,n
a2,1 a2,2 a2, a2, a2,n
a3,1 a3,2 a3, a3, a3,n
a4,1 a4,2 a4, a4, a4,n
an,1 an,2 an, an, an,n
ija2,1 a2,2 a2, a2, a2,n
a1,1 a1,2 a1, a1, a1,n
a3,1 a3,2 a3, a3, a3,n
a4,1 a4,2 a4, a4, a4,n
an,1 an,2 an, an, an,n
i j
i j j i
A
B
i j j i
A 1 i 1 1 j
B
i j j i
A 1 i 1 1 j
B 1 j 1 1 i 1
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INTERCHANGING FIRST TWO ROWSBA
Sign on a1,i a2,j A1,i; 2,j
i j j i
A 1 i 1 1 j
B 1 j 1 1 i 1
i j j i
A 1 i 1 1 j 1 i 1 1 j 1
B 1 j 1 1 i 1
i j j i
A 1 i 1 1 j 1 i 1 1 j 1
B 1 j 1 1 i 1 1 j 1 1 i
a1,1 a1,2 a1, a1, a1,n
a2,1 a2,2 a2, a2, a2,n
a3,1 a3,2 a3, a3, a3,n
a4,1 a4,2 a4, a4, a4,n
an,1 an,2 an, an, an,n
ija2,1 a2,2 a2, a2, a2,n
a1,1 a1,2 a1, a1, a1,n
a3,1 a3,2 a3, a3, a3,n
a4,1 a4,2 a4, a4, a4,n
an,1 an,2 an, an, an,n
ijiji j
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SWAPPING OTHER ROWS:
Induction If the first row is not involved, use the
inductive hypothesis If the first row is to be swapped with
row k, Swap row k with row two Swap rows one and two Swap row k with row two
detB detA
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PROPOSED PRESENTATION OF THE DETERMINANT
Motivation: Signed Area/Volume/Hyper-volume spanned by the rows
Development Thread:1. Simple Cases2. Row operations3. Semi-formal definition4. Transition to Cofactor (Permutation)
Definition5. Properties
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BENEFITS OF A VOLUME-FIRST APPROACH Better motivation Multiple views Students can develop significant ideas
on their own: Active Learning Students can anticipate theoretical ideas Students are motivated to prove row
operation results
Thank you
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Associated materials may be obtained by contacting
Ray RosentraterWestmont College955 La Paz RdSanta Barbara, CA [email protected]