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

    Exercises

### Exercise

Determine explicitly the column rank and the row rank of the matrix
    ${\begin{pmatrix}3&2&6\\4&1&5\\6&-1&3\end{pmatrix}}.$
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

Determine the determinant of a plane rotation.

### Exercise

Compute the determinant of the matrix
    ${\begin{pmatrix}1+3{\mathrm {i} }&5-{\mathrm {i} }\\3-2{\mathrm {i} }&4+{\mathrm {i} }\end{pmatrix}}.$

### Exercise

Compute the determinant of the matrix
    ${\begin{pmatrix}1&3&5\\2&1&3\\8&7&4\end{pmatrix}}.$

### Exercise

Compute the determinant of the matrix
    ${\begin{pmatrix}1&3&9&0\\-1&0&5&2\\0&1&3&6\\-3&0&0&7\end{pmatrix}}.$

### 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 $3\times 3$-matrix.

### Exercise

Let $M$ be the following square matrix
    $M={\begin{pmatrix}A&B\\0&D\end{pmatrix}}\,,$
where $A$ and $D$ are square matrices. Prove that $\det M=\det A\cdot \det D$.

### Exercise *

Determine for which $x\in \mathbb {C}$ the matrix
    ${\begin{pmatrix}x^{2}+x&-x\\-x^{3}+2x^{2}+5x-1&x^{2}-x\end{pmatrix}}$
is invertible.

### Exercise

Use the image to convince yourself that, given two vectors $(x_{1},y_{1})$ and $(x_{2},y_{2})$, the determinant of the $2\times 2$-matrix defined by these vectors is equal (up to sign) to the area of the plane parallelogram spanned by the vectors.

### Exercise

Let $K$ be a field and $n\in \mathbb {N} _{+}$. Show that the determinant
    $\operatorname {Mat} _{n}(K)=(K^{n})^{n}\longrightarrow K,M\longmapsto \det M,$
fulfills (for arbitrary $k\in \{1,\ldots ,n\}$ and arbitrary $n-1$ vectors $v_{1},\ldots ,v_{k-1},v_{k+1},\ldots ,v_{n}\in K^{n}$, for $u\in K^{n}$ and for $s\in K$) the equality
    $\det {\begin{pmatrix}v_{1}\\\vdots \\v_{k-1}\\su\\v_{k+1}\\\vdots \\v_{n}\end{pmatrix}}=s\det {\begin{pmatrix}v_{1}\\\vdots \\v_{k-1}\\u\\v_{k+1}\\\vdots \\v_{n}\end{pmatrix}}\,.$

### Exercise

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

### Exercise

Let $K$ be a field and $m,n,p\in \mathbb {N}$. Prove that the transpose of a matrix satisfy the following properties (where $A,B\in \operatorname {Mat} _{m\times n}(K)$, $C\in \operatorname {Mat} _{n\times p}(K)$ and $s\in K$).
    ${({A^{\text{tr}}})^{\text{tr}}}=A\,.$
    ${(A+B)^{\text{tr}}}={A^{\text{tr}}}+{B^{\text{tr}}}\,.$
    ${(sA)^{\text{tr}}}=s\cdot {A^{\text{tr}}}\,.$
    ${(A\circ C)^{\text{tr}}}={C^{\text{tr}}}\circ {A^{\text{tr}}}\,.$

### Exercise

Compute the determinant of the matrix
    ${\begin{pmatrix}0&2&7\\1&4&5\\6&0&3\end{pmatrix}},$
by expanding the matrix along every column and along every row.

### Exercise

Compute the determinant of all the $3\times 3$-matrices, such that in each column and in each row there are exactly one $1$ and two $0$s.

### Exercise

Let $z\in \mathbb {C}$ and let
    $\mathbb {C} \longrightarrow \mathbb {C} ,w\longmapsto zw,$
$\mathbb {R} ^{2}\rightarrow \mathbb {R} ^{2}$.
The next exercises use the following definition.
Let $V$ be a vector space over a field $K$. For $a\in K$, the linear mapping
    $\varphi \colon V\longrightarrow V,v\longmapsto av,$
is called homothety (or dilation)
$a$

### 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
    $A={\begin{pmatrix}5&7\\2&-4\end{pmatrix}}\,\,{\text{ and }}\,\,B={\begin{pmatrix}-3&1\\6&5\end{pmatrix}}.$

### Exercise

Confirm the Multiplication theorem for determinants for the matrices
    $A={\begin{pmatrix}1&4&1\\1&2&0\\0&1&1\end{pmatrix}}\,\,{\text{ and }}\,\,B={\begin{pmatrix}2&0&1\\0&1&0\\1&0&1\end{pmatrix}}.$
    Hand-in-exercises

### Exercise (m+ marks)

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
    $\operatorname {rk} \,\varphi =\operatorname {rk} \,M\,.$

### Exercise (3 marks)

Compute the determinant of the matrix
    ${\begin{pmatrix}1+{\mathrm {i} }&3-2{\mathrm {i} }&5\\{\mathrm {i} }&1&3-{\mathrm {i} }\\2{\mathrm {i} }&-4-{\mathrm {i} }&2+{\mathrm {i} }\end{pmatrix}}.$

### Exercise (3 marks)

Compute the determinant of the matrix
    $A={\begin{pmatrix}2&1&0&-2\\1&3&3&-1\\3&2&4&-3\\2&-2&2&3\end{pmatrix}}\,.$

### Exercise (4 marks)

Check the multiplication theorem for the determinants of the following matrices
    $A={\begin{pmatrix}3&4&7\\2&0&-1\\1&3&4\end{pmatrix}}\,\,{\text{ and }}\,\,B={\begin{pmatrix}-2&1&0\\2&3&5\\2&0&-3\end{pmatrix}}.$