# Mathematics for Applied Sciences (Osnabrück 2023-2024)/Part I/Exercise sheet 25

*Exercises*

The telephone companies and compete for a market, where the market customers in a year are given by the customers-tuple (where is the number of customers of in the year etc.). There are customers passing from one provider to another one during a year.

- The customers of remain for with while of them goes to and the same percentage goes to .
- The customers of remain for with while of them goes to and goes to .
- The customers of remain for with while of them goes to and goes to .

a) Determine the linear map (i.e. the matrix), which expresses the customers-tuple with respect to .

b) Which customers-tuple arises from the customers-tuple within one year?

c) Which customers-tuple arises from the customers-tuple in four years?

The newspapers and sell subscriptions, and they compete in a local market with customers. Within a year, one can observe the following movements.

- The subscribers of stick with a percentage of to , switch to , switch to and become nonreaders.
- The subscribers of stick with a percentage of to , switch to , switch to and become nonreaders.
- The subscribers of stick with a percentage of to , nobody switches to , switch to and become nonreaders.
- Among the nonreaders, subscribe to or , the rest remains nonreaders.

a) Establish the matrix, which describes the movement of customers within a year.

b) In a certain year, each of the three newspapers has subscribers and there are nonreaders. How does the distribution look like after a year?

c) The three newspapers expand to another city, where there are no newspapers at all so far, but also potential customers. How many subscribers does each newspaper have (and how many nonreaders) after three years, if the same movements hold in the new city?

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 is surjective if and only if the columns of the matrix form a system of generators for .

Let be an -matrix and the corresponding linear mapping. Show that is surjective if and only if there exists an -matrix such that .

Let

a) Show

b) Determine the inverse matrix of .

c) Solve the equation

Determine the inverse matrix of

Determine the inverse matrix of

Determine the inverse matrix of

Determine the inverse matrix of the complex matrix

a) Determine if the complex matrix

is invertible.

b) Find a solution to the inhomogeneous linear system of equations

Determine the inverse matrix of

Prove that the matrix

for all is the inverse of itself.

Let be a field. We denote by the
-matrix,
with entry at the position and entry everywhere else. Then the following matrices are called *elementary matrices*.

- .

Let be a field and a -matrix with entries in . Prove that the multiplication by the elementary matrices from the left with M has the following effects.

- exchange of the -th and the -th row of .
- multiplication of the -th row of by .
- addition of -times the -th row of to the -th row ().

Describe what happens when a matrix is multiplied from the right by an elementary matrix.

Prove that the elementary matrices are invertible. What are the inverse matrices of the elementary matrices?

Show that the shear matrix

may be written as a matrix product ,where and are diagonal matrices and is a shear matrix of the form .

Let

Find elementary matrices such that is the identity matrix.

*Hand-in-exercises*

===Exercise (6 (3+1+2) marks) === An animal population consists of babies (first year), freshers (second year), Halbstarke (third year), mature ones (fourth year) and veterans (fifth year), these animals can not become older. The total stock of these animals in a given year is given by a -tuple .

During a year of the babies become freshers, of the freshers become Halbstarke, of the Halbstarken become mature ones and of the mature ones reach the fifth year.

Babies and freshes can not reproduce yet, then they reach the sexual maturity and Halbstarke generate new pets and of the mature ones generate new babies, and the babies are born one year later.

a) Determine the linear map (i.e. the matrix), which expresses the total stock with respect to the stock .

b) What will happen to the stock in the next year?

c) What will happen to the stock in five years?

### Exercise (3 marks)

Let be a complex number and let

be the multiplication map, which is a -linear map. How does the matrix related to this map with respect to the real basis and look like? Let and be complex numbers with corresponding real matrices and . Prove that the matrix product is the real matrix corresponding to .

### Exercise (3 marks)

Compute the inverse matrix of

### Exercise (3 marks)

Perform the procedure to find the inverse matrix of the matrix

under the assumption that .

### Exercise (3 marks)

Let

Find elementary matrices such that is the identity matrix.

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