Mathematics for Applied Sciences (Osnabrück 2023-2024)/Part I/Exercise sheet 26/refcontrol

Exercises

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Determine explicitly the column rank and the row rank of the matrix

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

Describe linear dependencies (if they exist) between the rows and between the columns of the matrix.

===Exercise Exercise 26.2

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Show that the elementary operations on the rows do not change the column rank.

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Determine the determinantMDLD/determinant of a plane rotation.MDLD/plane rotation

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Compute the determinant of the matrix

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

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Compute the determinant of the matrix

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

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Compute the determinantMDLD/determinant of the matrixMDLD/matrix

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

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Prove by induction that the determinant of an upper triangular matrix is equal to the product of the diagonal elements.

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Check the multi-linearity and the property to be alternating, directly for the determinant of a ${}3\times 3$-matrix.

===Exercise Exercise 26.9

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Let ${}M$ be the following square matrix

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

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

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

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

is invertible.

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

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

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

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

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

===Exercise Exercise 26.13

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

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

1. ${}{({A^{\text{tr}}})^{\text{tr}}}=A\,.$

2. ${}{(A+B)^{\text{tr}}}={A^{\text{tr}}}+{B^{\text{tr}}}\,.$

3. ${}{(sA)^{\text{tr}}}=s\cdot {A^{\text{tr}}}\,.$

4. ${}{(A\circ C)^{\text{tr}}}={C^{\text{tr}}}\circ {A^{\text{tr}}}\,.$

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

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

$\mathbb {C} \longrightarrow \mathbb {C} ,w\longmapsto zw,$

be the associated multiplication. Compute the determinant of this map, considering it as a real-linear map

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

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 scale factor ${}a$.

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What is the determinant of a homothety?

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Check the multiplication theorem for determinants of two homotheties on a finite-dimensional vector space.

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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

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

Hand-in-exercises

===Exercise (m+ marks) Exercise 26.22

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

$\varphi \colon V\longrightarrow W$

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

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

Exercise (3 marks) Create referencenumber

Compute the determinant of the matrix

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

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Compute the determinant of the matrix

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

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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