# 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

${\displaystyle {\begin{pmatrix}3&2&6\\4&1&5\\6&-1&3\end{pmatrix}}.}$

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

${\displaystyle {\begin{pmatrix}1+3{\mathrm {i} }&5-{\mathrm {i} }\\3-2{\mathrm {i} }&4+{\mathrm {i} }\end{pmatrix}}.}$

### Exercise

Compute the determinant of the matrix

${\displaystyle {\begin{pmatrix}1&3&5\\2&1&3\\8&7&4\end{pmatrix}}.}$

### Exercise

Compute the determinant of the matrix

${\displaystyle {\begin{pmatrix}1&3&9&0\\-1&0&5&2\\0&1&3&6\\-3&0&0&7\end{pmatrix}}.}$

### 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 ${\displaystyle {}3\times 3}$-matrix.

### Exercise

Let ${\displaystyle {}M}$ be the following square matrix

${\displaystyle {}M={\begin{pmatrix}A&B\\0&D\end{pmatrix}}\,,}$

where ${\displaystyle {}A}$ and ${\displaystyle {}D}$ are square matrices. Prove that ${\displaystyle {}\det M=\det A\cdot \det D}$.

### Exercise *

Determine for which ${\displaystyle {}x\in \mathbb {C} }$ the matrix

${\displaystyle {\begin{pmatrix}x^{2}+x&-x\\-x^{3}+2x^{2}+5x-1&x^{2}-x\end{pmatrix}}}$

is invertible.

### Exercise

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

### Exercise

Let ${\displaystyle {}K}$ be a field and ${\displaystyle {}n\in \mathbb {N} _{+}}$. Show that the determinant

${\displaystyle \operatorname {Mat} _{n}(K)=(K^{n})^{n}\longrightarrow K,M\longmapsto \det M,}$

fulfills (for arbitrary ${\displaystyle {}k\in \{1,\ldots ,n\}}$ and arbitrary ${\displaystyle {}n-1}$ vectors ${\displaystyle {}v_{1},\ldots ,v_{k-1},v_{k+1},\ldots ,v_{n}\in K^{n}}$, for ${\displaystyle {}u\in K^{n}}$ and for ${\displaystyle {}s\in K}$) the equality

${\displaystyle {}\det {\begin{pmatrix}v_{1}\\\vdots \\v_{k-1}\\su\\v_{k+1}\\\vdots \\v_{n}\end{pmatrix}}=s\det {\begin{pmatrix}v_{1}\\\vdots \\v_{k-1}\\u\\v_{k+1}\\\vdots \\v_{n}\end{pmatrix}}\,.}$

### Exercise

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

### Exercise

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

1. ${\displaystyle {}{({A^{\text{tr}}})^{\text{tr}}}=A\,.}$
2. ${\displaystyle {}{(A+B)^{\text{tr}}}={A^{\text{tr}}}+{B^{\text{tr}}}\,.}$
3. ${\displaystyle {}{(sA)^{\text{tr}}}=s\cdot {A^{\text{tr}}}\,.}$
4. ${\displaystyle {}{(A\circ C)^{\text{tr}}}={C^{\text{tr}}}\circ {A^{\text{tr}}}\,.}$

### Exercise

Compute the determinant of the matrix

${\displaystyle {\begin{pmatrix}0&2&7\\1&4&5\\6&0&3\end{pmatrix}},}$

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

### Exercise

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

### Exercise

Let ${\displaystyle {}z\in \mathbb {C} }$ and let

${\displaystyle \mathbb {C} \longrightarrow \mathbb {C} ,w\longmapsto zw,}$
be the associated multiplication. Compute the determinant of this map, considering it as a real-linear map

${\displaystyle {}\mathbb {R} ^{2}\rightarrow \mathbb {R} ^{2}}$.

The next exercises use the following definition.

Let ${\displaystyle {}V}$ be a vector space over a field ${\displaystyle {}K}$. For ${\displaystyle {}a\in K}$, the linear mapping

${\displaystyle \varphi \colon V\longrightarrow V,v\longmapsto av,}$

is called homothety (or dilation)

with scale factor ${\displaystyle {}a}$.

### 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

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

### Exercise

Confirm the Multiplication theorem for determinants for the matrices

${\displaystyle A={\begin{pmatrix}1&4&1\\1&2&0\\0&1&1\end{pmatrix}}\,\,{\text{ and }}\,\,B={\begin{pmatrix}2&0&1\\0&1&0\\1&0&1\end{pmatrix}}.}$

Hand-in-exercises

### Exercise (m+ marks)

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

${\displaystyle \varphi \colon V\longrightarrow W}$

be a linear map, described by the matrix ${\displaystyle {}M\in \operatorname {Mat} _{m\times n}(K)}$ with respect to two bases. Prove that

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

### Exercise (3 marks)

Compute the determinant of the matrix

${\displaystyle {\begin{pmatrix}1+{\mathrm {i} }&3-2{\mathrm {i} }&5\\{\mathrm {i} }&1&3-{\mathrm {i} }\\2{\mathrm {i} }&-4-{\mathrm {i} }&2+{\mathrm {i} }\end{pmatrix}}.}$

### Exercise (3 marks)

Compute the determinant of the matrix

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

### Exercise (4 marks)

Check the multiplication theorem for the determinants of the following matrices

${\displaystyle A={\begin{pmatrix}3&4&7\\2&0&-1\\1&3&4\end{pmatrix}}\,\,{\text{ and }}\,\,B={\begin{pmatrix}-2&1&0\\2&3&5\\2&0&-3\end{pmatrix}}.}$