INTERMEDIATE TEST MATHEMATICS for ECONOMIC APPLICATIONS 7/12/2018
I M 1) cos sincos sin
cos sin . The two square roots of are:
cos sin cos sin
with .
The two roots are:
cos sin and
cos sin .
I M 2) The characteristic polynomial of is
.
Matrix has the eingevalues , and , thus admits multiple eigenvalues if and only if or .
If , an is a vector
eigenvector associated to the eigenvalue
satisfying the condition
or, in system form:
, and so .
If , an is a vector
eigenvector associated to the eigenvalue
satisfying the condition
or, in system form:
, and so .
I M 3) From we get:
, so the system:
.
From we get:
. From
we get and so
and and and and
.
So we get
.
The dimension of the Image of the linear map is equal to the Rank of the matrix while the dimension of its Kernel is equal to , the difference between the dimen- sion of the domain of the linear map and the Rank of the matrix .
For the Rank of , note that , thus the matrix has not full rank while the determinant of the sub-matrix is different from zero.
So the Rank of is , the dimension of the Image of is and the dimension of the Kernel of is .
I M 4) The three vectors , and are linearly dependent if and only if the deter- minant of the matrix is equal to .
; thus the three vectors are linearly dependent if and only if . For we get . From we get:
. So, for we get:
.
I M 5) Matrices and are similar if .
.
So
.
