TASKS Quantitative Methods/Mathematicsfor Economic Applications AA. 2019/20

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TASKS Quantitative Methods/Mathematics for Economic Applications AA. 2019/20

Intermediate Test December 2019

I M 1) Given the polynomial equation B ^# # B  5 œ !, determine the value of for5 which this equation admits the solution B œ /¹^%³. Then calculate the square roots of the other solution.

I M 2) Given the matrix  œ determine the value of the parameter such5

" 5 ! 5 "  "

!  " "

 

that the matrix admits the eigenvalue - œ ! and, for this value of , find one modal matrix5 which diagonalizes .

I M 3) If in the basis •œ"ß "ß ! à "ß  "ß ! à #ß !ß "     the vector has coordinates—

"ß #ß  ", determine its coordinates in the basis –œ"ß "ß ! à "ß  "ß ! à !ß #ß "    . I M 4) Given a linear map 0 À‘^$ Ä‘^$, ˜œ —† , determine the matrix knowing that such linear map satisfies these three conditions:

a) 0 "ß #ß " œ "ß $ß $ à    b) "ß "ß ! − Ker 0 à

c) "ß "ß " is an eigenvector of corresponding to the eigenvalue  - œ ".

I M 5) Given the linear system , check, depending on the pa-





B  #B  #B  B œ "

$B  B  #B œ "

B  $B  7B  5 B œ !

" # $ %

" # %

" # $ %

rameters and , the existence and the number of its solutions.7 5 I Winter Exam Session 2020

I M 1) If D œ  # # $ / ¹^{% 3}  # "  $ / ¹^{$ 3}, calcolate D.

I M 2) Given the matrix  œ determine the two values of the parameter for5

" " "

" ! "

" 5 "

 

which the matrix admits multiple eigenvalues. For these values of , check if the correspon-5 ding matrix is diagonalizable or not.

I M 3) Consider the linear map 0 À‘^% Ä‘^%, ˜œ —† for which:

0 B ß B ß B ß B " # $ % œ B  $B à #B  #B à B  #B à #B  %B" # " # $ % $ %.

Determine the dimensions of the Kernel and of the Image of this linear map, and then find a basis for the Kernel.

I M 4) Given the matrix œ " # determine at least a matrix , different from , that is 

# "

 

similar to .

II M 1) Given 0 Bß C œ B C  BC  BC  ^# ^# and @œcosαßsinα , determine at least two va- lues of for which it results α H 0 "_@  ß " œ H 0 "_@ß@^#  ß " .

II M 2) Solve the problem .

Max/min u.c.:





 



0 Bß C œ B  C C Ÿ "  B

"  B Ÿ C

# #

#

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II M 3) Given the equation 0 Bß C œ B C  BC  #BC  #B  #C œ !  ^$ ^$ , satisfied at  "ß " , verify that an implicit function B Ä C B  can be defined with it and that such function has a stationary point. Then determine the nature of this stationary point.

II M 4) Given the vectors —œ BCß #  $C  and ˜œ B  %ß BC , determine if pairs Bß C exist for which the scalar product of the two vectors — ˜† œ 0 Bß C  is maximum or mini- mum.

II Winter Exam Session 2020

I M 1) Transform D œ "  $  "  $ in trigonometric form and then calculate D .

"  3 "  3

  _$

I M 2) Given the matrix œ and the vector —œ , determine the

" # " "

 " 7 # "

" " 5 "

   

values of and for which the vector 7 5 ˜œ —† is perpendicular to the vector !ß "ß  " and with modulus equal to #%.

I M 3) Given the matrix  œ determine the two values of the parameter for5

" ! "

" 5 "

" ! "

 

which the matrix admits multiple eigenvalues. For these values of , check if the correspon-5 ding matrix is diagonalizable or not.

I M 4) Given the basis for ‘^$À –œ"ß "ß ! à "ß !ß " à !ß "ß "    , find the coordinates of the vector ˜ œ !ß "ß  "  in such basis.

II M 1) Given the system    satisfied at P ,

   

0 Bß Cß D œ B /  C /  # / œ !

1 Bß Cß D œ B C  BD  #BCD œ ! œ "ß "ß "

C B D

# # !

verify that an implicit function B Ä Cß D  can be defined with it, and then calculate the first order derivatives of this function.

II M 2) Solve the problem .

Max/min u.c.:









 





0 Bß C œ B  C  B C  $B B !

C ! C Ÿ %  B

2 2

II M 3) Given 0 Bß C œ B C  and the vectors •œ "ß "  and –œ  "ß " , let and be@ A their unit vectors. If W@0 B ß C ! !œ# and WA0 B ß C ! !œ !, determine the coordinates of the point B ß C_! _!.

II M 4) Given the function 0 Bß C œ B  BC  C  ^# ^$, determine the nature of its stationary points.

I Additional Exam Session QMfEA 2020

I M 1) After having determined the complex number which is the solution of the equationD

D D

"  3  "  3 œ #3, calculate its cubic roots ^$ D.

I M 2) Given the matrix œ , since œ  " is an eigenvalue of , find

" " "

! 5 5  "

! ! 5

 

 ^#  -

the value of the real parameter , verify that is diagonalizable and then find a matrix that5  diagonalizes .

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I M 3) Since the linear system has the solution , calculate



B  B  B œ !  

7B  B  B œ ! B  B  5B œ !

"ß !ß  "

" # $

the values of the real parameters and and then find a basis for the space of the solutions7 5 of the system.

I M 4) Determine the vectors parallel to the vector — œ "ß  "ß #  and with modulus equal to '.

II M 1) Given the equation 0 Bß C œ /  ^{B C}^# ^# /^BC œ ! satisfied at the point  !ß ! , verify that an implicit function B Ä C B  can be defined with it, and then calculate the first order derivative of this function.

II M 2) Solve the problem Max min   .

s v

Î 0 Bß C œ B  #C  $D Þ Þ B  C  D œ "%^# ^# ^#

II M 3) Given 0 Bß C œ B C  and 1 Bß C œ B  C  ^# ^# determine for which values of parameter α it is W_@0 "ß " œ  W_@1 "ß " , with @œcosαßsinα.

II M 4) Given the function 0 Bß Cß D œ B CD   ^# ^$ logC  B  / ^CD, determine the gradient vector of the function at T œ "ß #ß #_!  .

I Additional Exam Session MfEA 2020

I M 1) After having determined the complex number which is the solution of the equationD

D D

"  3  "  3 œ ", calculate its square roots D.

I M 2) Given the matrix œ , since œ " is a multiple eigenvalue for ,

" ! !

" 5  " !

" 5 5

 

  -

find the values of the real parameter and check, for such values, if is diagonalizable or5  not.

I M 3) Since the linear system has the solution , calculate the



5B  #B  B œ #  

#B  5B  B œ #

$B  B  5B œ $

"ß "ß "

" # $

value of the real parameter and then find the number of the solutions of the system.5

I M 4) Determine all the vectors orthogonal to the vector —_" œ #ß  "ß "  and to the vector

—_# œ "ß "ß "  .

II M 1) Given the equation 0 Bß C œ /  ^BC /^BC œ ! satisfied at the point  !ß ! , verify that an implicit function B Ä C B  can be defined with it, and then calculate the first order derivative of this function.

II M 2) Solve the problem Max min   . s v

Î 0 Bß C œ B  C  D Þ Þ B  C  D œ $^# ^# ^#

II M 3) Given 0 Bß C œ B C  and 1 Bß C œ B  C  determine for which values of parameter α it is W_@0 "ß " œ  W_@1 "ß " , with @œcosαßsinα.

II M 4) Given the function 0 Bß Cß D œ B C D   ^{# $} logD  B  / ^CD, determine the gradient vector of the function at T œ "ß #ß #!  .

I Summer Exam Session 2020 I M 1) Calculate the cubic roots of the number D œ # 3.

"  3



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I M 2) Given the matrix  œ , determine, on varying the parameter , its ei-5

"  # !

#  $ !

$ " 5

 

genvalues and their multiplicity, then establishing if there are values of for which the given5 matrix is diagonalizable.

I M 3) Determine, on varying the parameters and , the dimensions of the Image and the7 5 Kernel of the linear map ‘^& Ä‘ , 0 — œ —† , with  œ .

" ! " ! "

" " " 7 5

! " ! " 7

3  

 

I M 4) Given the vectors —_" œ "ß #ß " , —_# œ "ß  "ß #  and —_$ œ #ß #ß 5 , determine the value of the parameter for which the three vectors are linearly dependent.5

II M 1) Solve the problem : Max/min .

u.c. :

  

 

0 Bß C œ B  C B  "  " Ÿ C Ÿ "

#

II M 2) Given the function 0 Bß C œ /  ^BC, find the directions @œcosαßsinα for which it results W_@0 !ß ! œ ! Þ 

II M 3) Given the function 0 Bß Cß D œ B  C  D  BC  ^# 2 ^# ^# 2 , check the nature of its stationary points.

II M 4) Given the equation 0 Bß C œ B /  ^BC C /^CB œ ! satisfied at the point  !ß ! , verify that an implicit function B Ä C B  can be defined with it, and then calculate the first order derivative of this function.

II Summer Exam Session 2020 I M 1) Calculate the square roots of the number D œ "  3.

"  3

I M 2) Determine an orthogonal matrix that diagonalizes the matrix  œ .

! " "

" ! "

" " !

 

I M 3) Determine, on varying the parameter , the dimensions of the Image and the Kernel of7 the linear map ‘^$ Ä‘ , 0 — œ —† , with  œ .

" " #

#  " #

" # 7

3  

 

I M 4) Determine the value of the parameter for which the vector 5 —œ "ß 5ß "  can be expressed as a linear combination of the vectors —_" œ "ß #ß  "  and —_# œ "ß "ß #  and then find the coefficients of such combination.

u.c.:

 0 Bß C œ B  B C 

%B  C Ÿ %

# 2

2 2

II M 2) Given the function 0 Bß C œ B C  , let and be the unit vectors of @ A •œ "ß "  and –œ "ß  " ; determine the point B ß C_! _! if W_@0 B ß C _! _!œ # and W_A0 B ß C _! _!œ !. II M 3) In a stationary point of a function of two variables 0 Bß C  the Hessian matrix is equal to ‡ œ 5 " . Determine, on varying the parameter , the nature of such5

" 5  #

 

stationary point.

II M 4) Given the system sen cos satisfied at the point

      

 

0 Bß Cß D œ BC  BD œ "

1 Bß Cß D œ B C  BD  DC œ "^{3 2} ³ ³

T œ !ß "ß "_!  , verify that with it we can define an implicit function D Ä Bß C  and then calculate its first derivatives at D œ ".

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II Autumn Exam Session 2020 I M 1) Calculate the cubic roots of the number D œ "  "

"  3 "  3 .

I M 2) Given the matrix  œ , determine the values of the parameter for5

5 ! 5

! # !

5 ! 5

 

which the matrix admits a multiple eigenvalue.

I M 3) Given the linear system , check existence and number of its





B  #B œ "

#B  B  5 B œ # B  #B  'B œ 5

" $

" # $

solutions on varying the parameter .5

I M 4) Determine the matrix  œ + + knowing that "ß  " is an eigenvector rela-

+ +

 ^"" ^"#  

#" ##

tive to the eigenvalue -œ ! and that  "ß " is an eigenvector relative to - œ #. II M 1) Solve the problem :

Max/min u.c.:









 





0 Bß C œ B  C  C C Ÿ "  #B

B ! C !

Þ

# #

II M 2) Given the function 0 Bß C œ B  BC  ^# ^#, and the unit vector @œcosαßsinα, cal-

culate W and W . And then calculate W when α 1.

@ # #

@ß@ @ß@

0 "ß " 0 "ß " 0 "ß " œ

      %

II M 3) Given the function 0 Bß C œ B  B C  C  ^# ^# ^# determine existence and nature of its stationary points.

II M 4) Given the equation 0 Bß C œ B /  ^BC C /BC œ !, verify if at P! œ !ß !  it is pos- sible to define an implicit function having as dependent variable of and then calculate theC B partial derivative of the first order of C B  at B œ !.

II Additional Exam Session 2020

I M 1) Calcolate " $ 3^$Þ

I M 2) Given the linear map 0 À Ä having matrix œ , determi-

"  " # "

! # " #

# ! 5 7

‘^% ‘^$ 

 

ne the value of the parameters and if the Kernel and the Image have the same dimension.5 7 For such values find a basis for the Kernel of the linear map generated by Þ

I M 3) Given the matrix  œ , find the value of the parameter knowing5

# #  "

$ !  #

5 # "

 

that the matrix admits the eigenvalue - œ  " and then determine, for this value of the parameter, if the matrix is diagonalizable or not.

I M 4) Find the coordinates of the vector in the basis — • œ"ß !ß ! ß "ß "ß ! ß "ß !ß "     if it has coordinates #ß !ß  " in the basis –œ"ß "ß ! ß "ß  "ß " ß "ß !ß "    .

u.c. :

 0 Bß C œ B  C  B Ÿ C Ÿ "

# #

#

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II M 2) Given the function 0 Bß Cß D œ B  BC  C  D  ^# ^# ^# determine the nature of its stationary point.

II M 3) Given the function 0 Bß C œ /  ^{B C}^# ^# and the unit vector @œcosαßsinα calculate its directional derivatives W^#_{@ @}0 !ß ! .

II M 4) Given the equation 0 Bß C œ B  sinC  CcosB œ !, verify if at P! œ !ß !  it is pos- sible to define an implicit function having as dependent variable of and then calculate theC B partial derivative of the first order of C B  at B œ !.