Sample Exam: Exam 1 keys from S2010 and F2009. See also S2009, exam 1.

More theorems on inversesTHEOREM. 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.

Methods for computing a determinantSarrus' rule, 2x2 and 3x3 cases. Four rules for determinantsTriangular 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 DeterminantsExamples: 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.

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)

Transpose matrix(A^T)^T = A (A + B)^T = A^T + B^T (AB)^T = B^T A^T

Further properties of the adjugate matrixComputing det(A) from A and adj(A) in 10 seconds Problems involving adj(A): examples from exams. Adjugate identity A adj(A) = adj(A) A = det(A) I 3x3 case: 6 ways to compute det(A) from A, adj(A). 3x3 case: the 6 cofactor expansions

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. 3.5-44 proof, revisited.

Intro to Ch4Def: 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.