ISCTE – IUL
Optimization
优化考试助攻代写 Part 1 (7.0 points) 1.Let A, B, C and D be matrices with real entries suchthat:A ∈ M3×3and |A| = 4;Bis a matrix in the row echelon form that is ···
Finance and Accounting, Management, Marketing Management
July 22nd, 2020
School Year 2019/2020, 2nd Semester
Special Season Exam Total Time: 2h15 (plus 15min of tolerance)
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Important observations 优化考试助攻代写 :
Theuse of calculators is not allowed and all electronic devices must be turned off for the whole duration of test.
Theuse of course materials or any other consultation materials is not allowed.
Allreplies must be properly justified and written on the sheets provided, and the staple must not be Red ink and pencils are not allowed.
Thelast sheet may be used for auxiliary computations, or, exceptionally, to answer questions, if extra space is allowed.
No questions will be answered.
Question Points:
Part 1 优化考试助攻代写
(7.0 points)
1.
Let A, B, C and D be matrices with real entries suchthat:
- A ∈ M3×3and |A| = 4;
- Bis a matrix in the row echelon form that is obtained from A by applying the following sequence of elementary row operations:
R1 ↔ R2 ; R1 → 1 R1 ; R3 → R3 − 3R2
- Cis the matrix corresponding to the quadratic form Q(x, y, z) = x2 − y2 − z2 + 2yz;
3.
4.
Part 2 优化考试助攻代写
(13.0 points)
In the questions of Part 2 you should present your reasoning clearly, indicating all necessary calculations and justifications.
1.
Consider the linear system with unknowns x, y and z,
where α and β are real parameters.
(0.5 pts.) (a) Writethe given system in the matrix equation form AX = B, where X = [x y z]T
is a column-vector containing the unknowns.
(1.5 pts.) (b) UsingGaussian elimination, classify the system as a function of the real parameters
α and β.
(0.5 pts.) (c) Usingthe previous paragraph, solve the system for α = β = 1.
(0.5 pts.) (d) Ifwe look at the parameter β as an additional unknown, we still obtain a linear Represent such system in the matrix equation form CY = D, where Y = [x y z β]T , that is, determine the rectangular matrix C and the column-vector D.
2.
Let A be a 3 × 3 matrix suchthat:
(0.5 pts.) (a) What is the value of det(A + 4I3)?
(0.5 pts.) (b) Justifythat λ = 0 is an eigenvalue of A and (0, 1, 0) is a corresponding
eigenvector.
(0.5 pts.) (c) Justify that A is diagonalizable.
(0.5 pts.) (d) Find an invertiblematrix P M3×3 and a diagonal matrix D M3×3 such that D = P −1AP .
(0.5 pts.) (e) Withoutperforming any calculation, explain how to use the matrices P and D from the previous paragraph to compute the power matrix A5.
3. 优化考试助攻代写
Let A be an invertible matrix of size n × n and let λ be an eigenvalue of A and v a corresponding eigenvector.
(0.5 pts.) (a)Prove that λ is also an eigenvalue of AT.
(0.5 pts.) (b)Provethat v is also an eigenvector of A−1 and indicate the corresponding
4.
Consider the real function of two realvariables
f (x, y) = x3 − 3xy2 + 6x2 − 6y2 + 2
(1.5 pts.) (a) Find the critical points of f and classify them.
(0.5 pts.) (b) Determinean equation of the tangent plane to the graph surface of f at the point
(1, 1, 0).
(0.5 pts.) (c)Determinethe direction of greatest rate of increase of f at the point (1, 1).
5. 优化考试助攻代写
Considerthe function f (x, y) = x2 − y2.
(0.5 pts.) (a) Justifythat f has absolute extrema when subject to the condition x2 + y2 = 1.
(1.5 pts.) (b) UsingLagrange Multipliers Method, determine the values of the absolute extrema of
f subject to x2 + y2 = 1.
(0.5 pts.) (c) Canthe existence of absolute extrema of f be guaranteed if no restrictions to the domain of f are imposed?