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)

with scaling factor ${}a$.

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 defines 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 surjectivity.

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

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

Exercise

Solve the 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.

Exercise

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)

The Sarrusminant of a ${}n\times n$-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 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.
6. Give an example for an invertible matrix, where the Sarrusminant equals ${}0$.
7. Give an example for a not-invertible matrix, where the Sarrusminant equals ${}1$.

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 (3 marks)

Solve the 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)

Let ${}V$ be a finite-dimensional vector space over the complex numbers ${}\mathbb {C}$, and let

$\varphi \colon V\longrightarrow V$

be a ${}\mathbb {C}$-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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