# 2250-2 7:30am Lecture Record Week 6 F2010

Last Modified: October 03, 2010, 10:05 MDT.    Today: September 24, 2018, 01:22 MDT.

## 27 Sep, 29 Sep, 30 Sep: Michal and Laura

Exam 1. Problems 4,5. Take the exam 27 Sep or 29 Sep in JWB 335 at 12:50pm, or 30 Sep in WEB 103 at 7:25am.
Sample Exam: Exam 1 keys from S2010 and F2009. See also S2009, exam 1.
HTML: Exam links for the past 5 years (17.7 K, html, 18 Dec 2010)

## 27 Sep: Determinants, Survey of Results. Section 3.6

```More theorems on inverses
THEOREM. The inverse of inverse(A) is A itself.
THEOREM. If C and D have inverses, then so does CD and
inverse(CD) = inverse(D) inverse(C).
```
```College Algebra Background:
College algebra determinant definition
Sarrus' rule for 2x2 and 3x3 matrices.
```
References for 3.6 determinant theory and Cramer's Rule
Slides: Determinants 2008 (167.7 K, pdf, 23 Sep 2010)
Manuscript: Determinants, Cramer's rule, Cayley-Hamilton (186.5 K, pdf, 09 Aug 2009)
```Methods for computing a determinant
Sarrus' rule, 2x2 and 3x3 cases.
Four rules for determinants
Triangular Rule (one-arrow Sarrus' Rule): The determinant of
a triangular matrix is the product of the diagonal elements.
Multiply rule: B=answer after mult(t,m), then |A| = (1/m) |B|
Swap rule: B=answer after swap(s,t), then |A| = (-1) |B|
Combo rule: B=answer after combo(s,t,c), then |A| = |B|
Results on Determinants
Examples: Computing det(A) easily. When does det(A)=0?
THEOREM. Determinant values for elementary matrices:
det(E)=1 for combo(s,t,c),
det(E)=m for mult(t,m),
det(E)=-1 for swap(s,t).
```
```Survey of Main theorems:
Computation by the 4 rules, cofactor expansion, hybrid methods.
Determinant product theorem det(AB)=det(A)det(B).
Cramer's Rule for solving Ax=b:
x1 = delta1/delta, ... , xn = deltan/delta
```

## 28 Sep: Cramers Rule, Adjugate formula. Section 3.6

```Lecture
Cofactor expansion of det(A).
minor(A,i,j)
checkerboard sign (-1)^{i+j}
cofactor(A,i,j)=(sign)minor(A,i,j)
Details for 3x3 and 4x4.
Hybrid methods to evaluate det(A).
How to use the 4 rules to compute det(A) for any size matrix.
Computing determinants of sizes 3x3, 4x4, 5x5 and higher.
Frame sequences and determinants.
Formula for det(A) in terms of swap and mult operations.
Special theorems for det(A)=0
a zero row or col
duplicates rows
proportional rows.
Elementary matrices
Determinant product rule for an elementary matrix
Cramer's rule.
How to form the matrix of cofactors and its transpose.
```
THEOREM. The 4 rules for computing any determinant can be compressed into two rules,
1. det(triangular matrix)=the product of the diagonal elements, and
2. det(EA)=det(E)det(A), where E is an elementary combo, swap or mult matrix.
```Determinant product theorem
det(AB)=det(A)det(B) for any two square matrices A,B
Proof details.
Example.
```

Slides: Determinants 2010 (167.7 K, pdf, 23 Sep 2010)
Manuscript: Determinants, Cramer's rule, Cayley-Hamilton (186.5 K, pdf, 09 Aug 2009)
html: Problem notes F2010 (4.6 K, html, 26 Nov 2010)

## 29 Sep: Problem session 3.4, 3.5, 3.6

```       Problem 3.5-60a and 60b.
How to discover the relation B_n = 2 B_{n-1} - B_{n-2}
Induction proof in 3.5-60b.

Problems 3.3-10,20 using maple
Problems 3.4-20,30,34,40
Problems 3.5-16,26,44
Problems 3.6-6,20,32,40,60
Some problem details appear already in the online problem notes.
The lecture added more details, and complete solutions in several cases.

Maple computation of det(A), invverse(A), adjoint(A)
```

## 1 Oct: Problem Session Ch 3. Introduction to Chapter 4. Vector Space. Section 4.1.

```Transpose matrix
(A^T)^T = A
(A + B)^T = A^T + B^T
(AB)^T = B^T A^T
```
```Further properties of the adjugate matrix
Computing det(A) from A and adj(A) in 10 seconds
Problems involving adj(A): examples from exams.
3x3 case: 6 ways to compute det(A) from A, adj(A).
3x3 case: the 6 cofactor expansions
```
```Problem Details
Exercises 3.4-34 and 3.4-40.
Cayley-Hamilton Theorem.
It is a famous result in linear algebra which is the basis for
solving systems of differential equations.
Discussion of the Cayley-Hamilton theorem [Exercise 3.4-29;
Manuscript: Determinants, Cramer's rule, Cayley-Hamilton (186.5 K, pdf, 09 Aug 2009)  Superposition proof
Problem 3.4-40 is the superposition principle for the
matrix equation Ax=b. It is the analog of the differential
equation relation y=y_h + y_p.
Web notes on the problems.
Problem 3.4-29 is used in Problem 3.4-30.
How to solve problem 3.4-30.
3.5-44 proof, revisited.
```
``` Intro to Ch4
Def: Vector==package of data items.
Vectors are not arrows.
The 8-Property Vector Toolkit
Def: vector space, subspace
Working set == subspace.
Data set == Vector space
Examples of vectors:
Digital photos,
Fourier coefficients,
Taylor coefficients,
Solutions to DE. Example: y=2exp(-x^2) for DE y'=-2xy, y(0)=2.
```