Mathematics for Applied Sciences (Osnabrück 2011-2012)/Part I/Exercise sheet 11
- Warm-up-exercises
Exercise
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.
Exercise
Show that the elementary operations on the rows do not change the column rank.
Exercise
Compute the determinant of the matrix
Exercise
Compute the determinant of the matrix
Exercise
Prove by induction that the determinant of an upper triangular matrix is equal to the product of the diagonal elements.
Exercise
Check the multi-linearity and the property to be alternating, directly for the determinant of a -matrix.
Exercise
Let be the following square matrix
where and are square matrices. Prove that .
Exercise *
Determine for which the matrix
is invertible.
Exercise
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.
Exercise
Prove that you can expand the determinant according to each row and each column.
Exercise
Let be a field and . Prove that the transpose of a matrix satisfy the following properties (where , and ).
Exercise
Compute the determinant of the matrix
by expanding the matrix along every column and along every row.
Exercise
Compute the determinant of all the -matrices, such that in each column and in each row there are exactly one and two s.
Exercise
Let and let
be the associated multiplication. Compute the determinant of this map, considering it as a real-linear map
.
Exercise
What is the determinant of a homothety?
Exercise
Check the multiplication theorem for determinants of two homotheties on a finite-dimensional vector space.
Exercise
Check the multiplication theorem for determinants of the following matrices
- Hand-in-exercises
Exercise (3 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 (2 marks)
Compute the determinant of the elementary matrices.
Exercise (4 marks)
Check the multiplication theorem for the determinants of the following matrices