Mathematics for Applied Sciences (Osnabrück 2011-2012)/Part I/Exercise sheet 11

Warm-up-exercises

Exercise

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

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

Exercise

Compute the determinant of the matrix

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

Exercise

Compute the determinant of the matrix

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

Exercise

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

Exercise

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

Exercise

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 ${}\det M=\det A\cdot \det D$.

Exercise *

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.

Exercise

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.

Exercise

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

Exercise

Let ${}K$ be a field, and ${}m,n,p\in \mathbb {N}$. Prove that the transpose of a matrix satisfies 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. ${}{(A\circ C)^{\text{tr}}}={C^{\text{tr}}}\circ {A^{\text{tr}}}\,.$

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

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.

Exercise

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

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

Exercise

What is the determinant of a homothety?

Exercise

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

Exercise

Check the multiplication theorem for determinants of the following matrices

$A={\begin{pmatrix}5&7\\2&-4\end{pmatrix}}\,\,{\text{ and }}\,\,B={\begin{pmatrix}-3&1\\6&5\end{pmatrix}}.$

Hand-in-exercises

Exercise (3 marks)

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)

Compute the determinant of the matrix

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

Exercise (3 marks)

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 (2 marks)

Compute the determinant of the elementary matrices.

Exercise (4 marks)

Check the multiplication theorem for the determinants of the following matrices

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