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Exercise 1: Modelling with Linear Algebra

On a clay tablet from Babylon, approx. 2000 BC, the following problem is stated:

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

A

Provide two equations with two unknowns that contain the information from 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

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

Determine $\,\mA\transp\,.$

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

A

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

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

Given the matrices

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

consider the matrix equation

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

Which form should the matrix $\,\mathbf X\,$ have for $\,(*)\,$ to be meaningful? Solve the equation using Gauss-Jordan elimination and compare with the result from question a).

C

Justify that $\,\mathbf A\,$ is invertible. Solve $\,(*)\,$ by isolating $\,\mathbf X\,$ on the left-hand-side of $\,(*)\,.$

Exercise 4: Regular Matrix, Inverse Matrix

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

A

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

B

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

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

A

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

B

State the general solution to the homogeneous linear system of equations that correspond to the given inhomogeneous system of equations.

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

C

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

D

Determine the general solution to the homogeneous linear system, that corresponds to the given inhomogeneous system.