Mathematics for Applied Sciences (Osnabrück 2023-2024)/Part I/Exercise sheet 26



Exercises

Determine explicitly the column rank and the row rank of the matrix

Describe linear dependencies (if they exist) between the rows and between the columns of the matrix.


Show that the elementary operations on the rows do not change the column rank.


Determine the determinant of a plane rotation.


Compute the determinant of the matrix


Compute the determinant of the matrix


Compute the determinant of the matrix


Prove by induction that the determinant of an upper triangular matrix is equal to the product of the diagonal elements.


Check the multilinearity and the property to be alternating, directly for the determinant of a -matrix.


Let be the following square matrix

where and are square matrices. Prove that .


Determine for which the matrix

is invertible.


Use the image to convince yourself that, given two vectors and , the determinant of the -matrix defined by these vectors is equal (up to sign) to the area of the plane parallelogram spanned by the vectors.


Let be a field and . Show that the determinant

fulfills (for arbitrary and arbitrary vectors , for and for ) the equality


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


Let be a field, and . Prove that the transpose of a matrix satisfies the following properties (where , , and ).


Compute the determinant of the matrix

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


Compute the determinant of all the -matrices, such that in each column and in each row, there are exactly one and two s.


Let and let

be the associated multiplication. Compute the determinant of this map, considering it as a real-linear map

.


The next exercises use the following definition.

Let be a vector space over a field . For , the linear mapping

is called homothety (or dilation)

with scaling factor .

What is the determinant of a homothety?


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


Check the multiplication theorem for determinants of the following matrices


Confirm the Multiplication theorem for determinants for the matrices




Hand-in-exercises

Exercise (m+ marks)

Let be a field, and let and be vector spaces over of dimensions and . Let

be a linear map, described by the matrix with respect to two bases. Prove that


Exercise (3 marks)

Compute the determinant of the matrix


Exercise (3 marks)

Compute the determinant of the matrix


Exercise (4 marks)

Check the multiplication theorem for the determinants of the following matrices



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