# Linear algebra (Osnabrück 2024-2025)/Part I/Exercise sheet 17

    Exercise for the break

### Exercise

Let $M$ be an invertible $n\times n$-matrix. Show
    $\det {\left(M^{-1}\right)}={\left(\det M\right)}^{-1}\,.$
    Exercises

### Exercise

Let $A$ be an $m\times n$-matrix, and $B$ be an $n\times k$-matrix, where the columns of $B$ are linearly dependent. Show that the columns of $A\circ B$ are also linearly dependent.

### 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
    $A={\begin{pmatrix}1&4&0\\2&0&5\\0&2&-1\end{pmatrix}}\,\,{\text{ and }}\,\,B={\begin{pmatrix}1&2&7\\0&3&6\\0&-2&-3\end{pmatrix}}.$
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.
The following exercises use the concept of a group homomorphism.
Let $(G,\circ ,e_{G})$ and $(H,\circ ,e_{H})$ denote groups. A mapping
    $\psi \colon G\longrightarrow H$
is called group homomorphism, if the equality
    $\psi (g\circ g')=\psi (g)\circ \psi (g')\,$
holds for all
$g,g'\in G$

### Exercise

Let $K$ be a field, and $n\in \mathbb {N} _{+}$. Show that the determinant
    $\operatorname {GL} _{n}\!{\left(K\right)}\longrightarrow (K\setminus \{0\},\cdot ,1),M\longmapsto \det M,$
is a surjective group homomorphism.

### Exercise

Let $K$ be a field, and let $n,m\in \mathbb {N} _{+}$ with $n\leq m$. Define an injective group homomorphism
    $\operatorname {GL} _{n}\!{\left(K\right)}\longrightarrow \operatorname {GL} _{m}\!{\left(K\right)}.$

### Exercise

We consider the matrix
    ${\begin{pmatrix}3&4\\1&2\end{pmatrix}}.$
Show that this matrix defiens a group homomorphism from $\mathbb {Q} ^{2}$ to $\mathbb {Q} ^{2}$, and from $\mathbb {Z} ^{2}$ to $\mathbb {Z} ^{2}$ as well. Study this group homomorphism with respect to injectivity and surjeytivity.

### Exercise

Let $M$ be an $n\times n$-matrix with entries in $\mathbb {Z}$, and let
    $\varphi _{M}\colon \mathbb {Z} ^{n}\longrightarrow \mathbb {Z} ^{n}$
denote the corresponding group homomorphism. Show that $\varphi _{M}$ is bijective if and only if the determinant of $M$ equals $1$ or equals $-1$.

### Exercise

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

### 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

Endomorphisms/Determinant multiplication theorem/Exercise

### Exercise

Cramer's rule/2x2/2/Exercise

### Exercise

Cramer's rule/3x3/Exercise
    Hand-in-exercises

### Exercise Sarrus-determinant/4x4/Properties/Exercise

Sarrus-determinant/4x4/Properties/Exercise

### Exercise (4 marks)

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

### Exercise Cramer's rule/Linear system/2 4 3/1 5 7/3 5 2/Is 3 3 4/Exercise

Cramer's rule/Linear system/2 4 3/1 5 7/3 5 2/Is 3 3 4/Exercise

### Exercise Complex endomorphism/Complex and real determinant/Exercise

Complex endomorphism/Complex and real determinant/Exercise