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

    Exercise for the break

### Exercise Create referencenumber

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

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

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Check the multiplication theorem for determinants of the following matrices
    $A={\begin{pmatrix}5&7\\2&-4\end{pmatrix}}\,\,{\text{ and }}\,\,B={\begin{pmatrix}-3&1\\6&5\end{pmatrix}}.$

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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 spaceMDLD/vector space over a fieldMDLD/field $K$. For $a\in K$, the linear mappingMDLD/linear mapping
    $\varphi \colon V\longrightarrow V,v\longmapsto av,$
is called homothety (or dilation)
$a$

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What is the determinant of a homothety?MDLD/homothety

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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.MDLD/groups A mappingMDLD/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
$g,g'\in G$

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

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

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We consider the matrix
    ${\begin{pmatrix}3&4\\1&2\end{pmatrix}}.$
Show that this matrix defines a group homomorphismMDLD/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 injectivityMDLD/injectivity and surjectivity.MDLD/surjectivity

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Let $M$ be an $n\times n$-matrixMDLD/matrix with entries in $\mathbb {Z}$, and let
    $\varphi _{M}\colon \mathbb {Z} ^{n}\longrightarrow \mathbb {Z} ^{n}$
denote the corresponding group homomorphism.MDLD/group homomorphism Show that $\varphi _{M}$ is bijectiveMDLD/bijective if and only if the determinantMDLD/determinant of $M$ equals $1$ or equals $-1$.
===Exercise Exercise 17.11
Prove that you can expand the determinant according to each row and each column.

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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.

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Let $V$ be a finite-dimensionalMDLD/finite-dimensional (vs) $K$-vector space,MDLD/vector space and let $\varphi ,\psi \colon V\rightarrow V$ denote linear mappings.MDLD/linear mappings Show $\det {\left(\varphi \circ \psi \right)}=\det \varphi \det \psi$.

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Solve the linear systemMDLD/linear system
    ${\begin{pmatrix}6&4\\5&2\end{pmatrix}}{\begin{pmatrix}x_{1}\\x_{2}\end{pmatrix}}={\begin{pmatrix}8\\3\end{pmatrix}}\,$
with Cramer's rule.

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We consider the matrix
    $A={\begin{pmatrix}1&0&2\\2&1&3\\1&0&1\end{pmatrix}}\in \operatorname {Mat} _{3}(\mathbb {R} )\,.$
Solve the linear system $Ax=(1,0,2)^{t}$ using Cramer's rule. (check first that we may apply this rule).
    Hand-in-exercises

### Exercise (8 (3+1+1+1+2) marks) Create referencenumber

The Sarrusminant of a $n\times n$-matrixMDLD/matrix is computed by repeating the first $n-1$ columns of the matrix in the same order behind the matrix, and then by adding up the $n$ products of the diagonals and subtracting the $n$ products of the antidiagonals. We restrict to the case $n=4$. That is, for a matrix
    $M={\begin{pmatrix}a&b&c&d\\e&f&g&h\\l&m&n&m\\q&r&s&t\end{pmatrix}}\,,$
we consider
    ${\begin{pmatrix}a&b&c&d&a&b&c\\e&f&g&h&e&f&g\\l&m&n&m&l&m&n\\q&r&s&t&q&r&s\end{pmatrix}}$
and the Sarrusminant is
    $\operatorname {Sar} (M)=afnt+bgpq+chlr+dems-qmgd-rnha-speb-tlfc\,.$
* Show that the mapping is multilinearMDLD/multilinear (in the rows of the matrix).
    $\Psi \colon \operatorname {Mat} _{4}(K)\longrightarrow K,M\longmapsto \operatorname {Sar} (M),$
* Show that, for $4\times 4$-matrices which contain a zero-row, Sarrusminant is $0$.
* Show that, for $4\times 4$-matrices which contain a zero-column, the Sarrusminant is $0$.
* Show that, for an upper triangular matrix, the Sarrusminant is the product of the diagonal elements.
* Show that the Sarrusminant is not alternating.MDLD/alternating
* Give an example for an invertible matrix,MDLD/invertible matrix where the Sarrusminant equals $0$.
* Give an example for a not-invertible matrix, where the Sarrusminant equals $1$.

### Exercise (4 marks) Create referencenumber

Check the multiplication theorem for the determinantsMDLD/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 (3 marks) Create referencenumber

Solve the linear systemMDLD/linear system (over $\mathbb {Q}$)
    ${\begin{matrix}2y&+4z&+3&=&3\\y&+5z&+7&=&3\\3y&+5z&+2&=&4\,.\end{matrix}}$
using Cramer's rule.

### Exercise (8 marks) Create referencenumber

Let $V$ be a finite-dimensionalMDLD/finite-dimensional (vs) vector spaceMDLD/vector space over the complex numbersMDLD/complex numbers $\mathbb {C}$, and let
    $\varphi \colon V\longrightarrow V$
be a $\mathbb {C}$-linear mapping.MDLD/linear mapping We consider $V$ also as a real vector space of double dimension. $\varphi$ is also a real-linear mapping, which we denote by $\psi$. Show that between the complex determinant and the real determinant, the relation
    $\vert {\det \varphi }\vert ^{2}=\det \psi \,$
holds.