INTERMEDIATE TEST
MATHEMATICS for ECONOMIC APPLICATIONS 04/12/2017
I M 1) Using trigonometric form: " 3 œ#cos 1 sen 1 œ# / and
% 3 % 31%
" $ 3 œ # œ # / " 3 œ # / œ #/
" $ 3 # / #
cos 1 sen 1 ; thus
$ 3 $ 3 3
3 3
1 1
1
$ 1 "#
%
$
and
" 3 # " "
" $ 3 œ / œ / œ œ
# % %
% %
3"#1 31$ cos 3 sen
$ $
1 1
œ " $ œ $
% " "
# # 3 ) ) 3 .
I M 2) The characteristic polynomial of is -
-
-
-
: œ œ
# # "
! 5 !
" # #
œ 5 # " œ 5 # " œ 5 % $ œ
" #
- - - - - - -
-
# #
œ 5 - - $- "; the matrix has multiple eigenvalue if 5 œ " or 5 œ $. For 5 œ " the multiple eigenvalue is - œ " with algebraic multiplicity equal to .#
The matrix equal to
" †ˆ œ "
" # "
! ! !
" # "
has rank , so the geometric multipli- city of - œ " is equal to and so for # 5 œ " the matrix is diagonalizable.
For 5 œ $ the multiple eigenvalue is - œ $ has algebraic multiplicity equal to but# the matrix
$ †ˆ œ #
" # "
! ! !
" # "
has rank equal to , so its geometric mul- tiplicity is equal to and so for " 5 œ $ the matrix is not diagonalizable.
I M 3) The dimension of the Image of a linear map is equal to the rank of the matrix as- sociated to the map; the matrix , using the elementary operations:
‘" Ç‘% ‘# Ç‘$ ‘% Ñ% 7 †‘"
" ! ! 5
! " ! !
! ! " !
! ! ! " 75
, , , can be reduced to
and easily we see that: Rank œ $
% if
if 57 œ ". 57 Á "
So if 57 œ " DimImm 0 œ $ and Dim Ker 0 œ " while DimImm 0 œ % and Dim Ker 0 œ ! if 57 Á ".
I M 4) An element — belongs to the Kernel of the linear map if 0 — œX †—œ or
" " # !
# " " !
$ # $ !
" ! " !
† œ
B B B
"
#
$
that in system form is:
B B B B B
B B B B $B B B
B B B B B B B
B B B B
" # $ # $
" # $ # $ # $
" # $ # $ " $
" $ " $
# Ê # Ê
# œ ! $ œ !
# œ ! œ ! œ $
$ $ œ ! ' œ ! œ
œ ! œ
.
Every element of the ernel is K and so a basis for
B B "
B B
B B "
œ œ B †
" $
# $
$ $
$
$ $
the Kernel is UO/<0 œ"ß $ß ".
I M 5) Given the matrix œ we must verify that it has two eigenvalues
" ! "
! " !
" ! "
equal to those of the matrix # œ † .
# œ † œ † œ
" ! " " ! " # ! #
! " ! ! " ! ! " !
" ! " " ! " # ! #
.
The characteristic polynomial of is:
: œ œ œ " œ
" ! "
! " !
" ! "
" "
" "
- -ˆ -
-
-
-
-
- œ " - " -# " œ " - - # #- œ-" - - # œ ! and so the three eigenvalues of are -" œ !, -# œ ", -$ œ #.
The characteristic polynomial of # is:
: œ œ œ " œ
# ! #
! " !
# ! #
# #
# #
#
- -ˆ -
-
-
-
-
-
#
œ " - # -# % œ " - - # %- œ-" - - % œ ! and so the three eigenvalues of # are -" œ !, -# œ ", -$ œ %.
The Matrices and # have two common eigenvalues: œ !- and - œ ".
