COMPITI DI MATHEMATICS for economic applications AA. 2011/12
Intermediate Test December 2011
I M 1) Compute $ " #3 " #3 .
$ 3 " 3
I M 2) Given the matrices œ and œ , verify that they
! " ! ! " !
" ! " " ! "
! " ! ! " !
have the same eigenvalues, then verify if the matrix † is a diagonalizable matrix, and finally find the relationship between † and † .
I M 3) Find at least one matrix for which œ " # and œ $ # are similar.
# " ! "
I M 4) Starting from —" œ "ß "ß " and —# œ "ß "ß ! built an orthonormal basis and find, in such a basis, the coordinates of the vector ˜ œ "ß "ß " .
I M 5) Consider a linear application ‘% Ä‘$, ˜œ —† for which:
0 B ß B ß B ß B " # $ % œ B #B B $B à #B B #B à %B 7 B 5 B" # $ % " $ % # $ %.
Determine, on varying the parameters and , the dimensions of Kernel and Rankspace for7 5 such linear application.
I Winter Exam Session 2012
I M 1) Equation B # 3 B % #3 B %3 œ !$ # has complex solutions. Find such$ solutions and compute square roots of the solution which is in the first quadrant of the plan.‚ I M 2) Check values for such that 5 —" œ "ß "ß " , —# œ "ß #ß 5 and —$ œ "ß "ß 5 are a base for ‘$. In how many ways, on varying , may the vector 5 ˜œ %ß %ß % be expressed as a linear combination of — —", # and —$ ?
I M 3) Given the matrix œ check, depending on the variation of the pa-
$ ! "
" # "
" 5 $
rameter , existence of multiple eigenvalues and if the matrix is diagonalizable with an ortho-5 gonal matrix.
I M 4) Investigate, depending on the variation of the parameter , existence and number of5 solutions of the linear system —† œ˜ if œ and ˜œ .
" $ 5 "
" " # "
" # 5 "
II M 1) Solve the problem Max/min .
s.v.:
0 Bß C œ B C " B %C Ÿ "
# $
# #
II M 2) Given 0 Bß C œ C BC $ # and the unit vector of A "ß " , determine and plot in the
Bß C plan all the points B ß C! ! for which WA0 B ß C ! !œ !.
II M 3) Check if the function 0 Bß C œ B B C # is differentiable at !ß ! .
II M 4) Given the equation 0 Bß C œ B C / # $ BC œ ! satisfied at P! œ "ß " , verify that it defines an implicit function C œ C B , and determine the expression of the second degree Taylor's polynomial at B œ " for such implicit function.
II Winter Exam Session 2012
I M 1) Compute " 3$.
I M 2) Given œ , compute and check if and are dia-
" " "
! " "
! ! "
" " "
gonalizable matrices.
I M 3) Given œ , find the value for for which 5 - œ ! is an eigenvalue of ,
" " "
" ! #
" # 5
and find an orthogonal matrix which diagonalizes .
I M 4) Check for existence and number of solutions, on varying parameters and , for the7 5
linear system of equations .
B #B #B B œ !
#B $B B #B œ "
%B B $B 7B œ 5
" # $ %
" # $ %
" # $ %
II M 1) Given 1 À‘Ä‘#ß > Ä B ß B " # and 0 À‘# Ä‘$ß B ß B " #Ä C ß C ß C " # $, with chain rule compute ` C ß C ß C as a product of Jacobian matrices, and then apply such for-
` >
" # $
mula when 1 À > Ä sen>ßcos> and 0 À B ß B " #Ä B B ß B B ß B B " # " # " #.
II M 2) System log log satisfied at P defi-
0 Bß Cß D œ B C C D BC D œ !
1 Bß Cß D œ B C C D œ " œ !ß "ß "
$ #
$ # $ #
nes an implicit function B Ä C B ß D B ; find the equation of the tangent line to such func- tion at B œ !.
II M 3) Given 0 Bß Cß D œ B BD CD # # compute @0 "ß !ß " and # 0 "ß !ß " , where @
W W@ß@
is the unit vector of "ß "ß ".
II M 4) Solve the problem .
Max/min s.v.:
0 Bß C œ B C $B #C
#B C $ ! C Ÿ "
C B $
# #
I Additional Exam Session 2012
I M 1) Compute " 3) " 3 ). We recommend using trigonometric form of complex numbers.
I M 2) Check, depending on the variation of parameters and , existence and number of so-7 5 lutions of the linear system —† œ˜ if œ and ˜œ .
" ! " " #
" " # " "
" $ 7 5 "
I M 3) Given the matrix œ check, depending on the variation of the parame-
" ! 5
! " !
" ! "
ter , existence of multiple eigenvalues to check if the matrix is diagonalizable. Then find the5 value for such that 5 -œ ! is an eigenvalue for and an orthogonal matrix which diagonalizes .
I M 4) Vector — œ B ß B " # has coordinates "ß # in the base "ß " à "ß # . Determine its coordinates in the base #ß " à "ß ".
II M 1) Given 0 Bß C œ BC B C " # # , check for its local maximum and minimum points.
II M 2) Solve the problem . Max/min
s.t.:
0 Bß C œ B C
B C
% * Ÿ "
# #
# #
II M 3) Given the equation 0 Bß C œ B C #B C BC % œ ! $ # # $ which is satisfied at point P! œ "ß " , verify that it defines an implicit function C œ C B , and determine the expression of the second degree Taylor's polynomial at B œ " for such implicit function.
II M 4) Given 0 Bß C œ B BC C # #, and unit vectors of ? @ "ß " and "ß ", find the point B ß C! ! at which W?0 B ß C ! !œ# and W@0 B ß C ! !œ $#.
