Drill:College algebra determinant definition Sarrus' rule for 2x2 and 3x3 matrices.The Four RulesTriangular rule [one-arrow Sarrus' rule] combo rule swap rule mult rule 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:

x_{1}= delta_{1}/delta, ... , x_{n}= delta_{n}/delta Adjugate formula: A adj(A) = adj(A) A = det(A) I Adjugate inverse formula inverse(A) = adjugate(A)/det(A).

LectureCofactor 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. The adjugate matrix.

- det(triangular matrix)=the product of the diagonal elements, and
- det(EA)=det(E)det(A), where E is an elementary combo, swap or mult matrix.

Determinant product theoremdet(AB)=det(A)det(B) for any two square matrices A,B Proof details. Example.

Sample Exam: Exam 1 key from F2009. See also S2009, exam 1.

Problem DetailsExercises 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; see also Section 6.3]: Determinants, Cramer's rule, Cayley-Hamilton (186.5 K, pdf, 09 Aug 2009)ManuscriptSuperposition proofProblem 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. Problem 3.5-60a and 60b. How to discover the reation B_n = 2 B_{n-1} - B_{n-2} Induction proof in 3.5-60b.

Four Vector Models:Fixed vectors Triad i,j,k algebraic calculus model Physics and Engineering arrows Gibbs vectors.: vector models and vector spaces (110.3 K, pdf, 03 Oct 2009) Parallelogram law. Head minus tail rule.SlidesVector ToolkitThe 8-property toolkit for vectors. Vector spaces. Reading: Section 4.1 in Edwards-Penney, especially the 8 properties.Lecture: Abstract vector spaces.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. RGB color separation and matrix add Intensity adjustments and scalar multiply

- Digital photos and matrix add, scalar multiply visualization.

Web references for chapter 4. Repeated below in ch3-ch4 references.