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

$\left(\begin{array}{ccc}3& 2& 6\\ 4& 1& 5\\ 6& -1& 3\end{array}\right).$

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

$\left(\begin{array}{cc}1+3\mathrm{i}& 5-\mathrm{i}\\ 3-2\mathrm{i}& 4+\mathrm{i}\end{array}\right).$

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Compute the determinantMDLD/determinant of the matrix

$\left(\begin{array}{ccc}1& 3& 5\\ 2& 1& 3\\ 8& 7& 4\end{array}\right).$

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Compute the determinantMDLD/determinant of the matrixMDLD/matrix

$\left(\begin{array}{cccc}1& 3& 9& 0\\ -1& 0& 5& 2\\ 0& 1& 3& 6\\ -3& 0& 0& 7\end{array}\right).$

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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 multilinearity 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

$M=\left(\begin{array}{cc}A& B\\ 0& D\end{array}\right),$

where $A$ and $D$ are square matrices. Prove that $detM=detA\cdot detD$.

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Determine for which $x\in ℂ$ the matrix

$\left(\begin{array}{cc}{x}^2+x& -x\\ -x^3+2x^2+5x-1& x^2-x\end{array}\right)$

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

${\mathrm{Mat}}_n(K)=(K^n{)}^n⟶K,M⟼detM,$

fulfills (for arbitrary $k\in \{1,\dots ,n\}$ and arbitrary $n-1$ vectors $v_1,\dots ,v_{k-1},v_{k+1},\dots ,v_n\in K^n$, for $u\in K^n$ and for $s\in K$) the equality

$det\left(\begin{array}{c}{v}_1\\ ⋮\\ v_{k-1}\\ su\\ v_{k+1}\\ ⋮\\ v_n\end{array}\right)=sdet\left(\begin{array}{c}{v}_1\\ ⋮\\ v_{k-1}\\ u\\ v_{k+1}\\ ⋮\\ v_n\end{array}\right).$

===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,MDLD/field and $m,n,p\in \mathbb{N}$. Prove that the transpose of a matrix satisfies the following properties (where $A,B\in {\mathrm{Mat}}_{m\times n}(K)$, $C\in {\mathrm{Mat}}_{n\times p}(K)$, 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

$\left(\begin{array}{ccc}0& 2& 7\\ 1& 4& 5\\ 6& 0& 3\end{array}\right),$

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 $z\in ℂ$ and let

$ℂ⟶ℂ,w⟼zw,$

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

${ℝ}^2\to {ℝ}^2$.

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

$\phi :V⟶V,v⟼av,$

is called homothety (or dilation)

with the 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.

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Check the multiplication theorem for determinants of the following matrices

$A=\left(\begin{array}{cc}5& 7\\ 2& -4\end{array}\right)\text{ and }B=\left(\begin{array}{cc}-3& 1\\ 6& 5\end{array}\right).$

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Confirm the Multiplication theorem for determinants for the matrices

$A=\left(\begin{array}{ccc}1& 4& 1\\ 1& 2& 0\\ 0& 1& 1\end{array}\right)\text{ and }B=\left(\begin{array}{ccc}2& 0& 1\\ 0& 1& 0\\ 1& 0& 1\end{array}\right).$

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

$\phi :V⟶W$

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

$\mathrm{rk}\phi =\mathrm{rk}M.$

Exercise (3 marks) Create referencenumber

Compute the determinant of the matrix

$\left(\begin{array}{ccc}1+\mathrm{i}& 3-2\mathrm{i}& 5\\ \mathrm{i}& 1& 3-\mathrm{i}\\ 2\mathrm{i}& -4-\mathrm{i}& 2+\mathrm{i}\end{array}\right).$

Exercise (3 marks) Create referencenumber

Compute the determinant of the matrix

$A=\left(\begin{array}{cccc}2& 1& 0& -2\\ 1& 3& 3& -1\\ 3& 2& 4& -3\\ 2& -2& 2& 3\end{array}\right).$

Exercise (4 marks) Create referencenumber

Check the multiplication theorem for the determinantsMDLD/determinants of the following matrices

$A=\left(\begin{array}{ccc}3& 4& 7\\ 2& 0& -1\\ 1& 3& 4\end{array}\right)\text{ and }B=\left(\begin{array}{ccc}-2& 1& 0\\ 2& 3& 5\\ 2& 0& -3\end{array}\right).$

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