Linear algebra (Osnabrück 2024-2025)/Part I/Exercise sheet 17/refcontrol



Exercise for the break

Let be an invertibleMDLD/invertible (matrix) -matrix.MDLD/matrix Show




Exercises

Let be an -matrix,MDLD/matrix and be an -matrix, where the columns of are linearly dependent.MDLD/linearly dependent Show that the columns of are also linearly dependent.


Check the multiplication theorem for determinants of the following matrices


Confirm the multiplication theorem for determinants for the matrices


The next exercises use the following definition.

Let be a vector spaceMDLD/vector space over a fieldMDLD/field . For , the linear mappingMDLD/linear mapping

is called homothety (or dilation)

with scaling factor .

What is the determinant of a homothety?MDLD/homothety


Check the multiplication theorem for determinants of two homotheties on a finite-dimensional vector space.


The following exercises use the concept of a group homomorphism.

Let and denote groups.MDLD/groups A mappingMDLD/mapping

is called group homomorphism, if the equality

holds for all

.

Let be a field,MDLD/field and . Show that the determinantMDLD/determinant

is a surjectiveMDLD/surjective group homomorphism.MDLD/group homomorphism


Let be a field,MDLD/field and let with . Define an injectiveMDLD/injective group homomorphismMDLD/group homomorphism


We consider the matrix

Show that this matrix defines a group homomorphismMDLD/group homomorphism from to , and from to as well. Study this group homomorphism with respect to injectivityMDLD/injectivity and surjectivity.MDLD/surjectivity


Let be an -matrixMDLD/matrix with entries in , and let

denote the corresponding group homomorphism.MDLD/group homomorphism Show that is bijectiveMDLD/bijective if and only if the determinantMDLD/determinant of equals or equals .


===Exercise Exercise 17.11

change===

Prove that you can expand the determinant according to each row and each column.


Compute the determinant of the matrix

by expanding the matrix along every column and along every row.


Let be a finite-dimensionalMDLD/finite-dimensional (vs) -vector space,MDLD/vector space and let denote linear mappings.MDLD/linear mappings Show .


Solve the linear systemMDLD/linear system

with Cramer's rule.


We consider the matrix

Solve the linear system using Cramer's rule (check first that we may apply this rule).




Hand-in-exercises

Exercise (8 (3+1+1+1+2) marks) Create referencenumber

The Sarrusminant of a -matrixMDLD/matrix is computed by repeating the first columns of the matrix in the same order behind the matrix, and then by adding up the products of the diagonals and subtracting the products of the antidiagonals. We restrict to the case . That is, for a matrix

we consider

and the Sarrusminant is

  1. Show that the mapping

    is multilinearMDLD/multilinear (in the rows of the matrix).

  2. Show that, for -matrices that contain a zero-row, the Sarrusminant is .
  3. Show that, for -matrices that contain a zero-column, the Sarrusminant is .
  4. Show that, for an upper triangular matrix, the Sarrusminant is the product of the diagonal elements.
  5. Show that the Sarrusminant is not alternating.MDLD/alternating
  6. Give an example for an invertible matrix,MDLD/invertible matrix where the Sarrusminant equals .
  7. Give an example for a not-invertible matrix, where the Sarrusminant equals .


Check the multiplication theorem for the determinantsMDLD/determinants of the following matrices


Solve the linear systemMDLD/linear system (over )

using Cramer's rule.


Let be a finite-dimensionalMDLD/finite-dimensional (vs) vector spaceMDLD/vector space over the complex numbersMDLD/complex numbers , and let

be a -linear mapping.MDLD/linear mapping We consider also as a real vector space of double dimension. is also a real-linear mapping, which we denote by . Show that between the complex determinant and the real determinant, the relation

holds.



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