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

Exercise for the break

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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)

with scaling factor ${}a$.

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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

change===

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

${}\operatorname {Sar} (M)=afnt+bgpq+chlr+dems-qmgd-rnha-speb-tlfc\,.$

1. Show that the mapping

   $\Psi \colon \operatorname {Mat} _{4}(K)\longrightarrow K,M\longmapsto \operatorname {Sar} (M),$

   is multilinearMDLD/multilinear (in the rows of the matrix).

2. Show that, for ${}4\times 4$-matrices that contain a zero-row, the Sarrusminant is ${}0$.
3. Show that, for ${}4\times 4$-matrices that contain a zero-column, the Sarrusminant is ${}0$.
4. Show that, for an upper triangular matrix, the Sarrusminant is the product of the diagonal elements.
5. Show that the Sarrusminant is not alternating.MDLD/alternating
6. Give an example for an invertible matrix,MDLD/invertible matrix where the Sarrusminant equals ${}0$.
7. 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

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

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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.

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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.

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