\\\\( \nonumber \newcommand{\bevisslut}{$\blacksquare$} \newenvironment{matr}[1]{\hspace{-.8mm}\begin{bmatrix}\hspace{-1mm}\begin{array}{#1}}{\end{array}\hspace{-1mm}\end{bmatrix}\hspace{-.8mm}} \newcommand{\transp}{\hspace{-.6mm}^{\top}} \newcommand{\maengde}[2]{\left\lbrace \hspace{-1mm} \begin{array}{c|c} #1 & #2 \end{array} \hspace{-1mm} \right\rbrace} \newenvironment{eqnalign}[1]{\begin{equation}\begin{array}{#1}}{\end{array}\end{equation}} \newcommand{\eqnl}{} \newcommand{\matind}[3]{{_\mathrm{#1}\mathbf{#2}_\mathrm{#3}}} \newcommand{\vekind}[2]{{_\mathrm{#1}\mathbf{#2}}} \newcommand{\jac}[2]{{\mathrm{Jacobi}_\mathbf{#1} (#2)}} \newcommand{\diver}[2]{{\mathrm{div}\mathbf{#1} (#2)}} \newcommand{\rot}[1]{{\mathbf{rot}\mathbf{(#1)}}} \newcommand{\am}{\mathrm{am}} \newcommand{\gm}{\mathrm{gm}} \newcommand{\E}{\mathrm{E}} \newcommand{\Span}{\mathrm{span}} \newcommand{\mU}{\mathbf{U}} \newcommand{\mA}{\mathbf{A}} \newcommand{\mB}{\mathbf{B}} \newcommand{\mC}{\mathbf{C}} \newcommand{\mD}{\mathbf{D}} \newcommand{\mE}{\mathbf{E}} \newcommand{\mF}{\mathbf{F}} \newcommand{\mK}{\mathbf{K}} \newcommand{\mI}{\mathbf{I}} \newcommand{\mM}{\mathbf{M}} \newcommand{\mN}{\mathbf{N}} \newcommand{\mQ}{\mathbf{Q}} \newcommand{\mT}{\mathbf{T}} \newcommand{\mV}{\mathbf{V}} \newcommand{\mW}{\mathbf{W}} \newcommand{\mX}{\mathbf{X}} \newcommand{\ma}{\mathbf{a}} \newcommand{\mb}{\mathbf{b}} \newcommand{\mc}{\mathbf{c}} \newcommand{\md}{\mathbf{d}} \newcommand{\me}{\mathbf{e}} \newcommand{\mn}{\mathbf{n}} \newcommand{\mr}{\mathbf{r}} \newcommand{\mv}{\mathbf{v}} \newcommand{\mw}{\mathbf{w}} \newcommand{\mx}{\mathbf{x}} \newcommand{\mxb}{\mathbf{x_{bet}}} \newcommand{\my}{\mathbf{y}} \newcommand{\mz}{\mathbf{z}} \newcommand{\reel}{\mathbb{R}} \newcommand{\mL}{\bm{\Lambda}} \newcommand{\mnul}{\mathbf{0}} \newcommand{\trap}[1]{\mathrm{trap}(#1)} \newcommand{\Det}{\operatorname{Det}} \newcommand{\adj}{\operatorname{adj}} \newcommand{\Ar}{\operatorname{Areal}} \newcommand{\Vol}{\operatorname{Vol}} \newcommand{\Rum}{\operatorname{Rum}} \newcommand{\diag}{\operatorname{\bf{diag}}} \newcommand{\bidiag}{\operatorname{\bf{bidiag}}} \newcommand{\spanVec}[1]{\mathrm{span}{#1}} \newcommand{\Div}{\operatorname{Div}} \newcommand{\Rot}{\operatorname{\mathbf{Rot}}} \newcommand{\Jac}{\operatorname{Jacobi}} \newcommand{\Tan}{\operatorname{Tan}} \newcommand{\Ort}{\operatorname{Ort}} \newcommand{\Flux}{\operatorname{Flux}} \newcommand{\Cmass}{\operatorname{Cm}} \newcommand{\Imom}{\operatorname{Im}} \newcommand{\Pmom}{\operatorname{Pm}} \newcommand{\IS}{\operatorname{I}} \newcommand{\IIS}{\operatorname{II}} \newcommand{\IIIS}{\operatorname{III}} \newcommand{\Le}{\operatorname{L}} \newcommand{\app}{\operatorname{app}} \newcommand{\M}{\operatorname{M}} \newcommand{\re}{\mathrm{Re}} \newcommand{\im}{\mathrm{Im}} \newcommand{\compl}{\mathbb{C}} \newcommand{\e}{\mathrm{e}} \\\\)

Exercise 1: Modelling with Linear Algebra

On a clay tablet from Babylon, ca. 2000 BC the following problem is given:

A man, that is 30 years older than his son, will in 8 years become 4 times as old as the son. How old are the farther and the son?

A

State two equations in two unknowns that contain the information in the clay tablet. Reduce the augmented matrix corresponding to the system of equations and find in this way the solution.

B

Check that your solution fits the information on the clay tablet.

Exercise 2: Transposed System of Equations

A matrix is given by

$$\mA=\begin{matr}{rrrr} 1 & 3 & 2 & 4 \\\\ 3 & 7 & 2 & 8 \\\\ 2 & 4 & 0 & 4 \end{matr}$$
A

State $\,\mA\transp\,.$

%####### begin:question %Hvilke regneregler for transponerede matricer kender du? %####### end:question

B

Solve the matrix equation

$$\begin{matr}{rrr} x_1 & x_2 & x_3 \end{matr} \cdot \mA = \begin{matr}{rrrr} 2 & 5 & 2 & 6 \end{matr}\,.$$

Exercise 3: Systems of Equations Versus Matrix Equations

C

Solve using Gauss-Jordan-elimination the following linear systems of equations

$$ \begin{aligned} x_1 - x_2 &= -1\\\\ 2x_1 + x_2 &= 4 \end{aligned} $$
$$ \begin{aligned} y_1 - y_2 &= 3\\\\ 2y_1 + y_2 &= 0 \end{aligned} $$

Given the matrices

$$\,\mA=\begin{matr}{rr} 1 & -1 \\\\ 2 & 1\end{matr}\,\,\,\,\mathrm{and}\,\,\,\, \mB=\begin{matr}{rr} -1 & 3 \\\\ 4 & 0\end{matr}\,.$$

We consider the matrix equation

$$(*)\,\,\,\,\,\,\,\,\,\,\mA \mathbf X =\mB\,.$$
D

Which form should the matrix $\,\mathbf X\,$ have such that $\,(*)\,$ is meaningfull? Solve the equation using Gauss-Jordan elimination and compare with the result in question a).

E

Explain that $\,\mathbf A\,$ is invertible, and now solve $\,(*)\,$ by isolating $\,\mathbf X\,$ on the left-hand side in $\,(*)\,.$

Exercise 4: Regular Matrix, Inverse Matrix

Given the matrices \begin{equation} \mA = \begin{matr}{rr} 0 & 1 \\ 1 & 0 \\ 2 & 3 \end{matr} \quad \mathrm{and} \quad \mB = \begin{matr}{rrr} 3 & 2 & 1 \\ 0 & 1 & 2 \end{matr} \end{equation} The products $\,\mA\mB\,$ and $\,\mB\mA\,$ are square matrices and give rise to the following question.

F

By hand: Show that $ \mB\mA $ is regular, and determine $ (\mB\mA)^{-1} $.

G

By hand: Show that $ \mA\mB $ is singular (i.e. not regular), an therefore you cannot determine $ (\mA\mB)^{-1} $.

Exercise 5: The Structure of Solution Sets

We consider the system of equations \begin{equation} \begin{aligned} x_1 + x_2 + 2x_3 &= 3\
2x_1 - x_2 + 4x_3 &= 0\
x_1 + 3x_2 - 2x_3 &= 3\
-3x_1 - 2x_2 + x_3 &= 0\
\end{aligned} \end{equation}

H

Find the reduced row echelon form of the augmented matrix corresponding to the system of equations. From this determine the rank of both the coefficient matrix and the augmented matrix. How many solutions does the system have. State the solutions.

I

State the complete solution to the homogeneous linear system of equations, that corresponds to the given inhomogeneous system of equations.

Now we consider the system of equations \begin{equation} \begin{aligned} x_1 + 2x_2 + 3x_3 + 4x_4 + 5x_5 &= 1\
2x_1 + 3x_2 + 4x_3 + 5x_4 + x_5 &= 2\
3x_1 + 4x_2 + 5x_3 + 6x_4 - 3x_5 &= 3\
\end{aligned} \end{equation}

J

Find the reduced row echelon form of the augmented matrix corresponding to the system of equations. From this determine the rank of both the coefficient matrix and the augmented matrix. How many solutions does the system have. State these in standard parametric form.

K

State the complete solution to the homogeneous system of linear equations, that corresponds to the given inhomogeneous system of equations.