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Chapter 7: Problem 9

The monthly sales (in thousands) of burgers \((\mathrm{B} 1)\) and bites \((\mathrm{B} 2)\) in three fast-food restaurants \((R 1, R 2, R 3)\) are as follows: \begin{tabular}{l|ccc} & R1 & R2 & R3 \\ \hline B1 & 35 & 27 & 13 \\ B2 & 42 & 39 & 24 \end{tabular} \begin{tabular}{l|ccc} & January \\ & R1 & R2 & R3 \\ \hline B1 & 31 & 17 & 3 \\ B2 & 25 & 29 & 16 \end{tabular} (a) Write down two \(2 \times 3\) matrices \(\mathbf{J}\) and \(\mathrm{F}\) representing sales in January and February respectively. (b) By finding \(\boldsymbol{J}+\mathrm{F}\), write down the matrix for the total sales over the two months. (c) By finding \(J-F\), write down the matrix for the difference in sales for the two months.

Short Answer

Expert verified
Matrix \mathbf{J} for January: \begin{pmatrix} 31 & 17 & 3 \ 25 & 29 & 16 \end{pmatrix}. Matrix \mathrm{F} for February: \begin{pmatrix} 35 & 27 & 13 \ 42 & 39 & 24 \end{pmatrix}. Total sales: \begin{pmatrix} 66 & 44 & 16 \ 67 & 68 & 40 \end{pmatrix}. Difference: \begin{pmatrix} -4 & -10 & -10 \ -17 & -10 & -8 \end{pmatrix}.

Step by step solution

01

Identify Given Data

Extract the sales data for burgers (B1) and bites (B2) from the given tables. For January, Burger sales are [31, 17, 3] and Bite sales are [25, 29, 16]. For February, Burger sales are [35, 27, 13] and Bite sales are [42, 39, 24].
02

Formulate Matrix \mathbf{J}

Construct matrix \mathbf{J} representing sales in January as follows:\[\mathbf{J} = \begin{pmatrix}31 & 17 & 3 \25 & 29 & 16\end{pmatrix}\]
03

Formulate Matrix \mathrm{F}

Construct matrix \mathrm{F} representing sales in February as follows:\[\mathrm{F} = \begin{pmatrix}35 & 27 & 13 \42 & 39 & 24\end{pmatrix}\]
04

Calculate Total Sales Matrix \mathbf{J} + \mathrm{F}

Add the corresponding elements of matrices \mathbf{J} and \mathrm{F} to find the total sales for both months:\[\mathbf{J} + \mathrm{F} = \begin{pmatrix}31 & 17 & 3 \25 & 29 & 16\end{pmatrix} + \begin{pmatrix}35 & 27 & 13 \42 & 39 & 24\end{pmatrix} = \begin{pmatrix}66 & 44 & 16 \67 & 68 & 40\end{pmatrix}\]
05

Calculate Difference in Sales Matrix \mathbf{J} - \mathrm{F}

Subtract the corresponding elements of matrix \mathrm{F} from \mathbf{J} to find the difference in sales between the two months:\[\mathbf{J} - \mathrm{F} = \begin{pmatrix}31 & 17 & 3 \25 & 29 & 16\end{pmatrix} - \begin{pmatrix}35 & 27 & 13 \42 & 39 & 24\end{pmatrix} = \begin{pmatrix}-4 & -10 & -10 \-17 & -10 & -8\end{pmatrix}\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Matrix Addition
Matrix addition is a fundamental operation in linear algebra. It involves adding corresponding elements of two matrices to form a new matrix. In this exercise, the matrices \(\mathbf{J}\) and \(\mathrm{F}\) represent sales data for January and February respectively.

Each element in \(\mathbf{J}\) corresponds to a sales figure in a particular restaurant for January, and similarly for \(\mathrm{F}\) in February. The task is to find the total sales over both months.

To add these matrices, simply add the sales figures for each corresponding location:
\[\begin{pmatrix} 31 & 17 & 3 \ 25 & 29 & 16 \end{pmatrix} + \begin{pmatrix} 35 & 27 & 13 \ 42 & 39 & 24 \end{pmatrix} \]

This results in:
\[\begin{pmatrix} 66 & 44 & 16 \ 67 & 68 & 40 \end{pmatrix} \]

Thus, this matrix represents the combined sales data for January and February for each restaurant.
Matrix Subtraction
Matrix subtraction is another key operation in linear algebra. It involves subtracting elements of one matrix from the corresponding elements of another matrix to form a new matrix. In this exercise, we determine the difference in sales between January and February.

Using the matrices \(\mathbf{J}\) and \(\mathrm{F}\) again, subtraction is performed as follows:

Subtract the sales figures for each corresponding restaurant:
\[\begin{pmatrix} 31 & 17 & 3 \ 25 & 29 & 16 \end{pmatrix} - \begin{pmatrix} 35 & 27 & 13 \ 42 & 39 & 24 \end{pmatrix} \]

This results in:
\[\begin{pmatrix} -4 & -10 & -10 \ -17 & -10 & -8 \end{pmatrix} \]

This matrix represents the difference in sales between January and February for each restaurant. This operation helps in understanding how sales figures change over time.
Economic Data Analysis
Matrix operations such as addition and subtraction are crucial tools in economic data analysis. They allow analysts to combine and compare different sets of data efficiently.

In this exercise, the sales data for fast-food restaurants in January and February are analyzed using matrices.

By performing matrix addition, we combined the sales figures to get a complete picture of the sales performance over two months. This can help in identifying sales trends and making informed business decisions, such as inventory management, marketing strategies, and staffing needs.

Matrix subtraction, on the other hand, helps in pinpointing the exact differences in sales between the two months. This can highlight whether certain products or locations are performing better or worse and why that might be happening.

Using these matrix operations makes it easier to manage and analyze large sets of data, thus providing clear insights into economic patterns and aiding better decision-making.

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Most popular questions from this chapter

Evaluate the matrix product Ax, where $$ \mathbf{A}=\left[\begin{array}{lll} 1 & 4 & 7 \\ 2 & 6 & 5 \\ 8 & 9 & 5 \end{array}\right] \text { and } \quad \mathbf{x}=\left[\begin{array}{l} x \\ y \\ z \end{array}\right] $$ Hence show that the system of linear equations $$ \begin{array}{r} x+4 y+7 z=-3 \\ 2 x+6 y+5 z=10 \\ 8 x+9 y+5 z=1 \end{array} $$ can be written as \(A x=b\) where $$ \mathbf{b}=\left[\begin{array}{r} -3 \\ 10 \\ 1 \end{array}\right] $$

Use Cramer's rule to solve (a) \(\left[\begin{array}{rr}4 & -1 \\ -2 & 5\end{array}\right]\left[\begin{array}{l}x_{1} \\\ x_{2}\end{array}\right]=\left[\begin{array}{r}13 \\ 7\end{array}\right]\) for \(x_{1}\) (b) \(\left[\begin{array}{rrr}3 & 2 & -2 \\ 4 & 3 & 3 \\ 2 & -1 & 1\end{array}\right]\left[\begin{array}{l}x_{1} \\ x_{2} \\\ x_{3}\end{array}\right]=\left[\begin{array}{r}-15 \\ 17 \\\ -1\end{array}\right]\) for \(x_{2}\) (c) \(\left[\begin{array}{rrrr}1 & 0 & 2 & 3 \\ -1 & 5 & 4 & 1 \\ 0 & 7 & -3 & 6 \\ 2 & 4 & 5 & 1\end{array}\right]\left[\begin{array}{l}x_{1} \\ x_{2} \\\ x_{3} \\ x_{4}\end{array}\right]=\left[\begin{array}{r}-1 \\ 1 \\ -24 \\\ 15\end{array}\right]\) for \(x_{4}\)

Let $$ \mathbf{A}=\left[\begin{array}{rrr} 2 & 1 & 1 \\ 5 & 1 & 0 \\ -1 & 1 & 4 \end{array}\right] \quad \mathbf{B}=\left[\begin{array}{l} 1 \\ 2 \\ 1 \end{array}\right] \quad \mathbf{C}=\left[\begin{array}{ll} 1 & 2 \\ 3 & 1 \end{array}\right] \quad \mathbf{D}=\left[\begin{array}{rr} 1 & 1 \\ -1 & 1 \\ 2 & 1 \end{array}\right] \quad \text { and } \quad \mathbf{E}=\left[\begin{array}{lll} 1 & 2 & 3 \\ 4 & 5 & 6 \end{array}\right] $$ Find (where possible) (a) \(\mathrm{AB}\) (b) \(\mathbf{B A}\) (c) CD (d) \(\mathrm{DC}\) (e) \(\mathbf{A E}\) (f) EA (g) DE (h) ED

Each unit of water output requires inputs of \(0.1\) units of steel and \(0.2\) units of electricity. Each unit of steel output requires inputs of \(0.1\) units of water and \(0.2\) units of electricity. Each unit of electricity output requires inputs of \(0.2\) units of water and \(0.1\) units of steel. (a) Determine the level of total output needed to satisfy a final demand of 750 units of water, 300 units of steel and 700 units of electricity. (b) Write down the multiplier for water output due to changes in final demand for electricity. Hence calculate the change in water output due to a 100 unit increase in final demand for electricity.

(a) Solve the system of equations $$ \begin{aligned} &2 x_{1}+4 x_{2}=16 \\ &3 x_{1}-5 x_{2}=-9 \end{aligned} $$ using Cramer's rule to find \(x_{2}\). (b) Solve the system of equations $$ \begin{array}{r} 4 x_{1}+x_{2}+3 x_{3}=8 \\ -2 x_{1}+5 x_{2}+x_{3}=4 \\ 3 x_{1}+2 x_{2}+4 x_{3}=9 \end{array} $$ using Cramer's rule to find \(x_{3}\).
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