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

Exercises

Exercise

The telephone companies ${}A,B$ and ${}C$ compete for a market, where the market customers in a year ${}j$ are given by the customers-tuple ${}K_{j}=(a_{j},b_{j},c_{j})$ (where ${}a_{j}$ is the number of customers of ${}A$ in the year ${}j$ etc.). There are customers passing from one provider to another one during a year.

1. The customers of ${}A$ remain for ${}80\%$ with ${}A$, while ${}10\%$ of them goes to ${}B$, and the same percentage goes to ${}C$.
2. The customers of ${}B$ remain for ${}70\%$ with ${}B$, while ${}10\%$ of them goes to ${}A$, and ${}20\%$ goes to ${}C$.
3. The customers of ${}C$ remain for ${}50\%$ with ${}C$, while ${}20\%$ of them goes to ${}A$, and ${}30\%$ goes to ${}B$.

a) Determine the linear map (i.e. the matrix), which expresses the customers-tuple ${}K_{j+1}$ with respect to ${}K_{j}$.

b) Which customers-tuple arises from the customers-tuple ${}(12000,10000,8000)$ within one year?

c) Which customers-tuple arises from the customers-tuple ${}(10000,0,0)$ in four years?

Exercise

The newspapers ${}A,B$ and ${}C$ sell subscriptions, and they compete in a local market with ${}100000$ customers. Within a year, one can observe the following movements.

1. The subscribers of ${}A$ stick with a percentage of ${}80\%$ to ${}A$, ${}10\%$ switch to ${}B$, ${}5\%$ switch to ${}C$ and ${}5\%$ become nonreaders.
2. The subscribers of ${}B$ stick with a percentage of ${}60\%$ to ${}B$, ${}10\%$ switch to ${}A$, ${}20\%$ switch to ${}C$ and ${}10\%$ become nonreaders.
3. The subscribers of ${}C$ stick with a percentage of ${}70\%$ to ${}C$, nobody switches to ${}A$, ${}10\%$ switch to ${}B$ and ${}20\%$ become nonreaders.
4. Among the nonreaders, ${}10\%$ subscribe to ${}A,B$ or ${}C$, 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 ${}20000$ subscribers and there are ${}40000$ 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 ${}100000$ 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?

Exercise *

Let ${}K$ be a field and let ${}V$ and ${}W$ be vector spaces over ${}K$ of dimensions ${}n$ and ${}m$. Let

$\varphi \colon V\longrightarrow W$

be a linear map, described by the matrix ${}M\in \operatorname {Mat} _{m\times n}(K)$ with respect to two bases. Prove that ${}\varphi$ is surjective if and only if the columns of the matrix form a system of generators for ${}K^{m}$.

Exercise

Let ${}M$ be an ${}m\times n$-matrix and ${}\varphi \colon K^{n}\rightarrow K^{m}$ the corresponding linear mapping. Show that ${}\varphi$ is surjective if and only if there exists an ${}n\times m$-matrix ${}A$ such that ${}M\circ A=E_{m}$.

Exercise

Let

${}M={\begin{pmatrix}11&-20\\6&-11\end{pmatrix}}\,.$

a) Show

${}M^{2}=E_{2}\,.$

b) Determine the inverse matrix of ${}M$.

c) Solve the equation

${}M{\begin{pmatrix}x\\y\end{pmatrix}}={\begin{pmatrix}4\\-9\end{pmatrix}}\,.$

Exercise

Determine the inverse matrix of

${\begin{pmatrix}-{\frac {9}{4}}&0&\cdots &\cdots &0\\0&{\frac {50}{3}}&0&\cdots &0\\\vdots &\ddots &-{\frac {5}{3}}&\ddots &\vdots \\0&\cdots &0&10^{7}&0\\0&\cdots &\cdots &0&{\frac {2}{11}}\end{pmatrix}}.$

Exercise

Determine the inverse matrix of

${}M={\begin{pmatrix}2&7\\-4&9\end{pmatrix}}\,$

Exercise

Determine the inverse matrix of

${}M={\begin{pmatrix}1&2&3\\6&-1&-2\\0&3&7\end{pmatrix}}\,.$

Exercise

Determine the inverse matrix of the complex matrix

${}M={\begin{pmatrix}2+3{\mathrm {i} }&1-{\mathrm {i} }\\5-4{\mathrm {i} }&6-2{\mathrm {i} }\end{pmatrix}}\,.$

Exercise *

a) Determine if the complex matrix

${}M={\begin{pmatrix}2+5{\mathrm {i} }&1-2{\mathrm {i} }\\3-4{\mathrm {i} }&6-2{\mathrm {i} }\end{pmatrix}}\,$

is invertible.

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

${}M{\begin{pmatrix}z_{1}\\z_{2}\end{pmatrix}}={\begin{pmatrix}54+72{\mathrm {i} }\\0\end{pmatrix}}\,.$

Exercise

Determine the inverse matrix of

${\begin{pmatrix}0&0&\cdots &0&1\\0&0&\cdots &1&0\\\vdots &\vdots &1&\vdots &\vdots \\0&1&\cdots &0&0\\1&0&\cdots &0&0\end{pmatrix}}.$

Exercise

Prove that the matrix

${\begin{pmatrix}0&0&k+2&k+1\\0&0&k+1&k\\-k&k+1&0&0\\k+1&-(k+2)&0&0\end{pmatrix}}$

for all ${}k$ is the inverse of itself.

Let ${}K$ be a field. We denote by ${}B_{ij}$ the ${}n\times n$-matrix, with entry ${}1$ at the position ${}(i,j)$ and entry ${}0$ everywhere else. Then the following matrices are called elementary matrices.

1. ${}V_{ij}:=E_{n}-B_{ii}-B_{jj}+B_{ij}+B_{ji}$.
2. ${}S_{k}(s):=E_{n}+(s-1)B_{kk}{\text{ for }}s\neq 0$
3. ${}A_{ij}(a):=E_{n}+aB_{ij}{\text{ for }}i\neq j{\text{ and }}a\in K$

Exercise

Let ${}K$ be a field and ${}M$ a ${}n\times n$-matrix with entries in ${}K$. Prove that the multiplication by the elementary matrices from the left with M has the following effects.

1. ${}V_{ij}\circ M=$ exchange of the ${}i$-th and the ${}j$-th row of ${}M$.
2. ${}(S_{k}(s))\circ M=$ multiplication of the ${}k$-th row of ${}M$ by ${}s$.
3. ${}(A_{ij}(a))\circ M=$ addition of ${}a$-times the ${}j$-th row of ${}M$ to the ${}i$-th row (${}i\neq j$).

Exercise

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

Exercise

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

Exercise

Show that the shear matrix

${}A_{ij}(a)=E_{n}+aB_{ij}\,$

may be written as a matrix product ${}M\circ N\circ L$,where ${}M$ and ${}L$ are diagonal matrices and ${}N$ is a shear matrix of the form ${}A_{ij}(1)$.

Exercise

Let

${}M={\begin{pmatrix}4&3\\5&1\end{pmatrix}}\,.$

Find elementary matrices ${}E_{1},\ldots ,E_{k}$ such that ${}E_{k}\circ \cdots \circ E_{1}\circ M$ is the identity matrix.

Hand-in-exercises

Exercise (6 (3+1+2) marks)

An animal population consists of babies (first year), freshers (second year), rockers (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 ${}j$ is given by a ${}5$-tuple ${}B_{j}=(b_{1,j},b_{2,j},b_{3,j},b_{4,j},b_{5,j})$.

During a year, ${}7/8$ of the babies become freshers, ${}9/10$ of the freshers become rockers, ${}5/6$ of the rockers become mature ones, and ${}2/3$ of the mature ones reach the fifth year.

Babies and freshers can not reproduce yet, then they reach sexual maturity, and ${}10$ rockers generate ${}5$ new pets, and ${}10$ of the mature ones generate ${}8$ new babies, and the babies are born one year later.

a) Determine the linear map (i.e., the matrix), which expresses the total stock ${}B_{j+1}$ with respect to the stock ${}B_{j}$.

b) What will happen to the stock ${}(200,150,100,100,50)$ in the next year?

c) What will happen to the stock ${}(0,0,100,0,0)$ in five years?

Exercise (3 marks)

Let ${}z\in \mathbb {C}$ be a complex number and let

$\mathbb {C} \longrightarrow \mathbb {C} ,w\longmapsto zw,$

be the multiplication map, which is a ${}\mathbb {C}$-linear map. How does the matrix related to this map with respect to the real basis ${}1$ and ${}{\mathrm {i} }$ look like? Let ${}z_{1}$ and ${}z_{2}$ be complex numbers with corresponding real matrices ${}M_{1}$ and ${}M_{2}$. Prove that the matrix product ${}M_{2}\circ M_{1}$ is the real matrix corresponding to ${}z_{1}z_{2}$.

Exercise (3 marks)

Compute the inverse matrix of

${}M={\begin{pmatrix}2&3&2\\5&0&4\\1&-2&3\end{pmatrix}}\,.$

Exercise (3 marks)

Perform the procedure to find the inverse matrix of the matrix

${\begin{pmatrix}a&b\\c&d\end{pmatrix}}$

under the assumption that ${}ad-bc\neq 0$.

Exercise (3 marks)

Let

${}M={\begin{pmatrix}4&6\\7&-3\end{pmatrix}}\,.$

Find elementary matrices ${}E_{1},\ldots ,E_{k}$ such that ${}E_{k}\circ \cdots \circ E_{1}\circ M$ is the identity matrix.

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