Linear algebra (Osnabrück 2024-2025)/Part I/Exercise sheet 12

Exercise for the break

Exercise

Prove that the elementary matrices are invertible. What are the inverse matrices of the elementary matrices?

Exercises

Exercise

Show that an invertible matrix ${}M$ neither has a zero row nor a zero column.

Exercise

Let ${}M$ be an ${}n\times n$-matrix such that there exist ${}n\times n$-matrices ${}A,B$ satisfying ${}M\circ A=E_{n}$ and ${}B\circ M=E_{n}$. Show that ${}A=B$ holds, and that ${}M$ is invertible.

Exercise

Let ${}M$ and ${}N$ be invertible ${}n\times n$-matrices. Show that also ${}M\circ N$ is invertible, and that

${}(M\circ N)^{-1}=N^{-1}\circ M^{-1}\,$

holds.

Exercise

Let ${}M$ denote an invertible upper triangular matrix. Show that the inverse matrix ${}M^{-1}$ is also an upper triangular matrix.

Exercise

Let ${}M$ denote an invertible upper triangular matrix. Show that the diagonal elements of ${}M$ are not ${}0$.

Exercise *

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 ${}\varphi$ is surjective if and only if the columns of the matrix form a system of generators for ${}K^{m}$.

Exercise

Let ${}K$ be a field and ${}M$ a ${}n\times n$-matrix with entries in ${}K$. Prove that the multiplication by the elementary matrices from the left with M has the following effects.

1. ${}V_{ij}\circ M=$ exchange of the ${}i$-th and the ${}j$-th row of ${}M$.
2. ${}(S_{k}(s))\circ M=$ multiplication of the ${}k$-th row of ${}M$ by ${}s$.
3. ${}(A_{ij}(a))\circ M=$ addition of ${}a$-times the ${}j$-th row of ${}M$ to the ${}i$-th row (${}i\neq j$).

Exercise

Describe what happens when a matrix is multiplied from the right by an elementary matrix.

Exercise

1. Transform the matrix equation

   ${}{\begin{pmatrix}3&7\\-4&5\end{pmatrix}}{\begin{pmatrix}x&y\\z&w\end{pmatrix}}={\begin{pmatrix}1&0\\0&1\end{pmatrix}}\,$

   into a system of linear equations.

2. Solve this linear system.

Exercise

Let ${}M={\begin{pmatrix}a&b\\c&d\end{pmatrix}}$ and ${}A={\begin{pmatrix}x&y\\z&w\end{pmatrix}}$ be matrices over a field ${}K$ such that

${}A\circ M={\begin{pmatrix}1&0\\0&1\end{pmatrix}}\,$

holds. Show directly that

${}M\circ A={\begin{pmatrix}1&0\\0&1\end{pmatrix}}\,$

holds as well.

Exercise

Determine the inverse matrix of

${}M={\begin{pmatrix}2&7\\-4&9\end{pmatrix}}\,$

Exercise

Determine the inverse matrix of

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

Exercise

Determine the inverse matrix of

${}M={\begin{pmatrix}1&2&3\\6&-1&-2\\0&3&7\end{pmatrix}}\,.$

Exercise

Determine the inverse matrix of the complex matrix

${}M={\begin{pmatrix}2+3{\mathrm {i} }&1-{\mathrm {i} }\\5-4{\mathrm {i} }&6-2{\mathrm {i} }\end{pmatrix}}\,.$

Exercise *

a) Determine if the complex matrix

${}M={\begin{pmatrix}2+5{\mathrm {i} }&1-2{\mathrm {i} }\\3-4{\mathrm {i} }&6-2{\mathrm {i} }\end{pmatrix}}\,$

is invertible.

b) Find a solution to the inhomogeneous linear system of equations

${}M{\begin{pmatrix}z_{1}\\z_{2}\end{pmatrix}}={\begin{pmatrix}54+72{\mathrm {i} }\\0\end{pmatrix}}\,.$

Exercise

Perform, for the matrix

${\begin{pmatrix}3&-2&5\\1&7&-4\\9&17&-2\end{pmatrix}},$

the inverting algorithm, until it is obvious that the matrix is not invertible.

Exercise

Let

${}M={\begin{pmatrix}4&3\\5&1\end{pmatrix}}\,.$

Find elementary matrices ${}E_{1},\ldots ,E_{k}$ such that ${}E_{k}\circ \cdots \circ E_{1}\circ M$ is the identity matrix.

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

Let ${}M$ be an ${}m\times n$-matrix and ${}\varphi \colon K^{n}\rightarrow K^{m}$ the corresponding linear mapping. Show that ${}\varphi$ is surjective if and only if there exists an ${}n\times m$-matrix ${}A$ such that ${}M\circ A=E_{m}$.

Exercise

Let ${}B$ be an ${}n\times p$-matrix, and let ${}A$ be an ${}m\times n$-matrix. Show that, for the column rank, the estimate

${}\operatorname {rk} \,A\circ B\leq {\min {\left(\operatorname {rk} \,A,\operatorname {rk} \,B,\right)}}\,$

holds.

Exercise

Let ${}M$ be an ${}m\times n$-matrix, and let ${}A$ be an invertible ${}m\times m$-matrix. Show that, for the column rank, the equality

${}\operatorname {rk} \,A\circ M=\operatorname {rk} \,M\,$

holds.

A block matrix is an ${}m\times n$-matrix of the form

${\begin{pmatrix}A&B\\C&D\end{pmatrix}},$

where ${}A$ is an ${}r\times s$-matrix, ${}B$ is an ${}r\times (n-s)$-matrix, ${}C$ is an ${}(m-r)\times s$-matrix and ${}D$ is an ${}(m-r)\times (n-s)$-matrix.

Exercise

Let a block matrix of the form

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

be given. Show that the rank of ${}M$ equals the sum of the ranks of ${}A$ and of ${}B$.

Hand-in-exercises

Exercise (3 marks)

Compute the inverse matrix of

${}M={\begin{pmatrix}2&3&2\\5&0&4\\1&-2&3\end{pmatrix}}\,.$

Exercise (4 marks)

Determine the inverse matrix of the complex matrix

${}M={\begin{pmatrix}5+8{\mathrm {i} }&3-7{\mathrm {i} }\\2-9{\mathrm {i} }&4-5{\mathrm {i} }\end{pmatrix}}\,.$

Exercise (4 marks)

Let

${}M={\begin{pmatrix}3&5&-1\\4&7&2\\2&-3&6\end{pmatrix}}\,.$

Find elementary matrices ${}E_{1},\ldots ,E_{k}$ such that ${}E_{k}\circ \cdots \circ E_{1}\circ M$ is the identity matrix.

Exercise ( marks)

Prove that the matrix

${\begin{pmatrix}0&0&k+2&k+1\\0&0&k+1&k\\-k&k+1&0&0\\k+1&-(k+2)&0&0\end{pmatrix}}$

for all ${}k$ is the inverse of itself.

Exercise (3 marks)

Perform the procedure to find the inverse matrix of the matrix

${\begin{pmatrix}a&b\\c&d\end{pmatrix}}$

under the assumption that ${}ad-bc\neq 0$.

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

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

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