14Week 49

Compute the products AB and BA, if possible, when A and B are, respectively:

1. $\textbf{A} = \begin{pmatrix} 0 & -2 \\ 3 & \phantom{-}1 \end{pmatrix}$ and $\textbf{B} = \begin{pmatrix} -1 & 4 \\ \phantom{-}1 & 5 \end{pmatrix}$
2. $\textbf{A} = \begin{pmatrix} 8 & 3 & -2 \\ 1 & 0 & \phantom{-}4 \end{pmatrix}$ and $\textbf{B} = \begin{pmatrix} 2 & -2 \\ 4 & \phantom{-}3 \\ 1 & -5 \end{pmatrix}$
3. $\textbf{A} = \begin{pmatrix} -1 & 0 \\ \phantom{-}2 & 4 \end{pmatrix}$ and $\textbf{B} = \begin{pmatrix} \phantom{-}3 & 1 \\ -1 & 1 \\ \phantom{-}0 & 2 \end{pmatrix}$
4. $\textbf{A} = \begin{pmatrix} \phantom{-} 0\\ -2 \\ \phantom{-}4 \end{pmatrix}$ and $\textbf{B} = \begin{pmatrix} 0 & -2 & 3 \end{pmatrix}$

Two companies, A and B, compete for market shares in the telecommunications market. In the beginning, A has a third of the market and B has two thirds. But Danes often change their provider, so over the next year, and each year after that, the following changes happen:

• A keeps 90% of customers and lose 10% to B
• B keeps 60% of customers and lose 40% to A

We can represent market shares for the two companies as a market share vector, which is a column vector $\mathbf{s}$. The definition of the market share vector is a column vector with non-negative elements that sum to 1. We define the matrix $\mathbf{T}$ and the initial market share vector, $\mathbf{s},$ as $$\displaystyle\textbf{T} = \begin{pmatrix} \displaystyle \frac{9}{10} & \displaystyle\frac{2}{5} \\[12pt] \displaystyle \frac{1}{10} & \displaystyle \frac{3}{5} \end{pmatrix}, \qquad \displaystyle\textbf{s} = \begin{pmatrix} \displaystyle \frac{1}{3} \\[12pt] \displaystyle \frac{2}{3} \end{pmatrix}$$ Note that each element of $\mathbf{T}$, $t_{ij}$, is equal to the fraction of $j$'s customers that become $i's$ customers in the following year. $\mathbf{T}$ is therefore called a transition matrix.

1. Show that $\mathbf{Ts}$ is a market share vector.
2. Calculate $\mathbf{T}^2\mathbf{s}$ and interpret this matrix product. Here $\mathbf{T}^2\mathbf{s}$ is the same as $\mathbf{TTs}$.

A country's economy has three important sectors, $x, y$ and $z$. To produce their output the sectors depend on input from the two other sectors and the sector itself. The size of these inputs and outputs, exchanged between the sectors, are shown in billions in the figure below, which does not include the part of the output which is exported or used outside the three sectors.

1. Construct a matrix $\mathbf{B}$ where rows and columns correspond to $x$, $y$ and $z$ and an element corresponds to how much the sector in the row contributes to the sector in the column.

Write the following system of equations in matrix notation: $$\begin{aligned} 2x_1 - 5x_2 &= 3 \\ 5x_1 +8x_2 &= 5 \end{aligned}$$

Write the following system of equations in matrix notation: $$\begin{aligned} x + y + z + t &= a \\ x + 3y + 2z + 4t &= b \\ x + 4y + 8z &= c \\ 2x + z - t &= d \end{aligned}$$

(Previous exam problem)Consider the following system of equations: $$\begin{aligned} 3x_1 - x_2 - x_3 &= 5 \\ 2x_1 +x_2 - 2x_3 &= 0 \\ x_1 + 3x_2 + x_3 &= 1 \end{aligned}$$

1. Determine A, x, and b, so that the system of equations can be written in matrix form as:
   $$\textbf{Ax}= \textbf{b}$$
2. Compute CA and AD, where
   $\textbf{C} = \begin{pmatrix} \phantom{-}7 & -2 & 3 \\ -4 & \phantom{-}4 & 4 \\ \phantom{-}5 & -10 & 5 \end{pmatrix}$ and $\textbf{D} = \begin{pmatrix} \phantom{-}5 & 24 & 28 \\ 20 & 12 & 24 \\ -5 & 20 & 0 \end{pmatrix}$

Previous exam problem.A company produces three products, A, B and C. The demand for the three products is described by functions of market parameters (constants) $k_A$, $k_B$ and $k_C$ as well as the prices of the three products $x_A$, $x_B$ and $x_C$. The demand functions are:

• Product A: $\quad k_A - 2x_A + x_C$
• Product B: $\quad k_B + 2x_A - 2x_B + 2x_C$
• Product C: $\quad k_C + x_A + x_B - 2x_C$

The supply of the three products is given by the following functions:

• Product A: $\quad 3x_A -6$
• Product B: $\quad x_B -9$
• Product C: $\quad 2x_C - 9$

1. Set up a system of equations expressing the equilibrium conditions in the market, i.e., when supply equals demand for each of the three products.
2. Determine $\mathbf{A}$, $\mathbf{x}$ and $\mathbf{b}$ so the system of equations can be written in matrix form as
   $$\mathbf{Ax} = \mathbf{b}$$

Two matrices A and B are given by: $\textbf{A} = \begin{pmatrix*}[r] 3 & -5 & 7 \\ -3 & 1 & 2 \\ 4 & -9 & 4 \end{pmatrix*}$ and $\textbf{B} = \begin{pmatrix*}[r] 2 & 1 & 0 \\ 3 & 2 & -2 \\ 0 & 2 & -2 \end{pmatrix*}.$

(Previous exam problem)Let $\textbf{A} = \begin{pmatrix} 5 & 3 \\ 0 & 5 \end{pmatrix}$, $\textbf{B} = \begin{pmatrix} -8 & 0 & 7 \\ \phantom{-}1 & 3 & 2 \end{pmatrix}$, and $\textbf{C} = \begin{pmatrix} 1 & 0 \\ 0 & 3 \\ 7 & 1 \end{pmatrix}$Show that $(\textbf{AB})\textbf{C} = \textbf{A}(\textbf{BC})$.