C. Ray Rosentrater Westmont College 2013 Joint Mathematics Meetings

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.

Motivation: Want to study a function with a matrix variable. Development Thread: 1. Permutations 2. Elementary Products (Definition) 3. Evaluation by Row Reduction (No justification) 4. Properties 5. Cofactor Expansion (No justifiction) 6. Application: Crammer’s Rule

Motivation: Another important number associated with a square matrix. Development Thread: 1. Permutations 2. Definition 3. Properties (Row ops & Evaluation via triangular matrices) 4. Computation via Cofactors (3x3 justified) 5. Applications: Crammer’s rule

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

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 testing 2. Cofactor Definition 3. Properties (Row operations, Product) 4. Applications: Crammer’s rule, Matrix codes, Cross product

Motivation: Singularity testing Development Thread: 1. 2x2, 3x3 singularity testing 2. Cofactor Definition 3. Properties: Row operations (not justified), Products, Transposes 4. Applications: Crammer’s rule, Volume, Transformations

Not amenable to active learning E. G. O.

Motivation: Signed Area/Volume/Hyper-volume of the parallelogram (etc.) spanned by the rows Development Thread: 1. Simple Cases 2. Row operations 3. Semi-formal definition & computational method 4. Transition to Cofactor (Permutation) Definition 5. Properties

Motivation: Signed Area/Volume/Hyper-volume spanned by the rows Development Thread: 1. Simple Cases 2. Row operations 3. Semi-formal definition 4. Transition to Cofactor (Permutation) Definition 5. Properties

Motivation: Signed Area/Volume/Hyper-volume spanned by the rows Development Thread: 1. Simple Cases 2. Row operations 3. Semi-formal definition & computation 4. Transition to Cofactor (Permutation) Definition 5. Properties

Determinant = signed “volume” of the parallelogram spanned by the rows  To Compute: Use row replacements to put in triangular form, multiply the diagonal entries

Motivation: Signed Area/Volume/Hyper-volume spanned by the rows Development Thread: 1. Simple Cases 2. Row operations 3. Semi-formal definition 4. Transition to Cofactor (Permutation) Definition 5. Properties

Why have another method? A motivating example

State the Cofactor definition Verify the definitions agree Check simple (diagonal) case Check row operation behavior

 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

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

BA

BA

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

Motivation: Signed Area/Volume/Hyper-volume spanned by the rows Development Thread: 1. Simple Cases 2. Row operations 3. Semi-formal definition 4. Transition to Cofactor (Permutation) Definition 5. Properties

 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

Associated materials may be obtained by contacting Ray Rosentrater Westmont College 955 La Paz Rd Santa Barbara, CA