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

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.

- 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.

Adjugate MatrixTranspose matrix and properties. (A^T)^T = A (A + B)^T = A^T + B^T (AB)^T = B^TA^T The adjugate matrix. How to form the matrix of cofactors and its transpose. Maple answer check.Proofs: Special theorems for det(A)=0a zero row or col duplicates rows proportional rows RREF(A) not the identityProblem DetailsExercises 3.4-30. Problem 3.4-29 is used in Problem 3.4-30. How to solve problem 3.4-30 without 3.4-29. Cayley-Hamilton Theorem. Result of 3.4-28,29. 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. Showed how to write up the proof of 3.4-40(a). 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. Exercises 3.4, 3.5, 3.6. Solution to 3.5-44.Introduction to Ch4Vector space. Subspace. Vectors Def: Vector==package of data items. Vectors are not arrows. Examples of vectors: Digital photos, Fourier coefficients, Taylor coefficients, Solutions to DE. Example: y=2exp(-x^2) for DE y'=-2xy, y(0)=2.