introducing the determinant

INTRODUCING THE DETERMINANT

C. Ray RosentraterWestmont College

2013 Joint Mathematics Meetings

rosentr@westmont.edu

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.

WHAT ARE THE COMMON APPROACHES?

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

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

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

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

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

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

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

SIMPLE CASES

a 0

0 b

a,b 0

a 0,b 0

a,b 0

deta 0

0 b ab

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

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

ROW OPERATIONS: ROW SCALING

v1

v1

v2

v2

sv1

sv1

v2v3

sv1v1

v2

v1

v2

v1

detB sdetA

ROW OPERATIONS: ROW SWAP

v2

v1v2

v1

v2v3

v1

v2

v3

v1

detB detA

ROW OPERATIONS: ROW REPLACEMENT

v2

v1

v2v3

v1

v2 + sv1 v3

v1

v2

detB detA

v2

v1

v1 + sv2v2 + sv1

v2

v1

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

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

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

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

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

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

TRANSITION TO COFACTOR DEFINITION

• State the Cofactor definition• Verify the definitions agree

• Check simple (diagonal) case• Check row operation behavior

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

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

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

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

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

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

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

rosentr@westmont.edu

Thank you

Associated materials may be obtained by contacting

Ray RosentraterWestmont College955 La Paz RdSanta Barbara, CA 93108805.565.6185rosentr@westmont.edu