Problem 52 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

The total output of loudspeaker systems of the Acrosonic Company at their three production facilities for May and June is given by the matrices \(A\) and \(B\), respectively, where The unit production costs and selling prices for these loudspeakers are given by matrices \(C\) and \(D\), respectively, where Compute the following matrices and explain the meaning of the entries in each matrix. a. \(A C\) b. \(A D\) c. \(B C\) d. \(B D\) e. \((A+B) C\) f. \((A+B) D\) g. \(A(D-C)\) h. \(B(D-C)\) i. \((A+B)(D-C)\)

Step-by-Step Solution

Answer

The matrix multiplications represent various combinations of total outputs, production costs, and selling prices for the Acrosonic Company in May and June. The calculations show the total production costs (AC, BC, and (A+B)C), total selling prices (AD, BD, and (A+B)D), and total profits (A(D-C), B(D-C), and (A+B)(D-C)) for each product type in May, June, and both months combined.

Step 1: Matrix Multiplication: AC

\( AC = \begin{bmatrix} a_{11} & a_{12} & a_{13}\\ a_{21} & a_{22} & a_{23} \end{bmatrix} \begin{bmatrix} c_{11}\\ c_{21}\\ c_{31} \end{bmatrix} \) b. Compute the matrix AD: The result of this calculation will be a 2x1 matrix representing the total selling price for each product type in May.

Step 2: Matrix Multiplication: AD

\( AD = \begin{bmatrix} a_{11} & a_{12} & a_{13}\\ a_{21} & a_{22} & a_{23} \end{bmatrix} \begin{bmatrix} d_{11}\\ d_{21}\\ d_{31} \end{bmatrix} \) c. Compute the matrix BC: The result of this calculation will be a 2x1 matrix representing the total production costs for each product type in June.

Step 3: Matrix Multiplication: BC

\( BC = \begin{bmatrix} b_{11} & b_{12} & b_{13}\\ b_{21} & b_{22} & b_{23} \end{bmatrix} \begin{bmatrix} c_{11}\\ c_{21}\\ c_{31} \end{bmatrix} \) d. Compute the matrix BD: The result of this calculation will be a 2x1 matrix representing the total selling price for each product type in June.

Step 4: Matrix Multiplication: BD

\( BD = \begin{bmatrix} b_{11} & b_{12} & b_{13}\\ b_{21} & b_{22} & b_{23} \end{bmatrix} \begin{bmatrix} d_{11}\\ d_{21}\\ d_{31} \end{bmatrix} \) e. Compute the matrix (A+B)C: The result of this calculation will be a 2x1 matrix representing the total production costs for each product type in both May and June.

Step 5: Calculating (A+B)C

\( (A+B)C = \left(\begin{bmatrix} a_{11} & a_{12} & a_{13}\\ a_{21} & a_{22} & a_{23} \end{bmatrix} + \begin{bmatrix} b_{11} & b_{12} & b_{13}\\ b_{21} & b_{22} & b_{23} \end{bmatrix}\right) \begin{bmatrix} c_{11}\\ c_{21}\\ c_{31} \end{bmatrix} \) f. Compute the matrix (A+B)D: The result of this calculation will be a 2x1 matrix representing the total selling price for each product type in both May and June.

Step 6: Calculating (A+B)D

\( (A+B)D = \left(\begin{bmatrix} a_{11} & a_{12} & a_{13}\\ a_{21} & a_{22} & a_{23} \end{bmatrix} + \begin{bmatrix} b_{11} & b_{12} & b_{13}\\ b_{21} & b_{22} & b_{23} \end{bmatrix}\right) \begin{bmatrix} d_{11}\\ d_{21}\\ d_{31} \end{bmatrix} \) g. Compute the matrix A(D-C): The result of this calculation will be a 2x1 matrix representing the total profit for each product type in May.

Step 7: Calculating A(D-C)

\( A(D-C) = \begin{bmatrix} a_{11} & a_{12} & a_{13}\\ a_{21} & a_{22} & a_{23} \end{bmatrix} \left(\begin{bmatrix} d_{11}\\ d_{21}\\ d_{31} \end{bmatrix} - \begin{bmatrix} c_{11}\\ c_{21}\\ c_{31} \end{bmatrix}\right) \) h. Compute the matrix B(D-C): The result of this calculation will be a 2x1 matrix representing the total profit for each product type in June.

Step 8: Calculating B(D-C)

\( B(D-C) = \begin{bmatrix} b_{11} & b_{12} & b_{13}\\ b_{21} & b_{22} & b_{23} \end{bmatrix} \left(\begin{bmatrix} d_{11}\\ d_{21}\\ d_{31} \end{bmatrix} - \begin{bmatrix} c_{11}\\ c_{21}\\ c_{31} \end{bmatrix}\right) \) i. Compute the matrix (A+B)(D-C): The result of this calculation will be a 2x1 matrix representing the total profit for each product type in both May and June.

Step 9: Calculating (A+B)(D-C)

\( (A+B)(D-C) = \left(\begin{bmatrix} a_{11} & a_{12} & a_{13}\\ a_{21} & a_{22} & a_{23} \end{bmatrix} + \begin{bmatrix} b_{11} & b_{12} & b_{13}\\ b_{21} & b_{22} & b_{23} \end{bmatrix}\right) \left(\begin{bmatrix} d_{11}\\ d_{21}\\ d_{31} \end{bmatrix} - \begin{bmatrix} c_{11}\\ c_{21}\\ c_{31} \end{bmatrix}\right) \)

Key Concepts

Matrix Multiplication

Matrix multiplication is a way to combine two matrices, resulting in a new matrix. It involves summing the products of rows of the first matrix and columns of the second. This operation is crucial when dealing with multiple data sets or variables, like in production data analysis. For instance, in the problem at hand, matrices represent quantities related to manufacturing costs and sales. Performing matrix multiplication helps to combine these matrices logically, allowing us to derive useful results such as total costs or revenues.

Understanding matrix multiplication involves:

• Selecting a row from the first matrix and a column from the second matrix.
• Multiplying corresponding entries and summing these products.
• Placing the resulting sum in the corresponding position of the new matrix.

When applied to production and financial data, this operation helps synthesize complex data into interpretable insights.

Production Costs

Production costs in matrix terms can be articulated by multiplying the outputs from a production matrix with a cost matrix. This combination gives a matrix that captures the total production costs for each product over a specific period, like May or June in this scenario. Understanding this relationship is vital for manufacturing businesses.

To break it down:

• The production matrix contains quantities produced for different product types across different facilities.
• The cost matrix represents the unit cost of manufacturing each product at these facilities.
• Matrix multiplication of these matrices outputs the total cost, assigning a separate result for each product type.

This calculation enables companies to efficiently track and manage expenses associated with production.

Selling Price

The selling price is another crucial concept calculated through matrix multiplication. By multiplying production matrices with the selling price matrix, companies can determine the projected sales revenue for their loudspeaker systems.

Here's how it works:

• Each entry in the production matrix represents the number of units of a product type.
• The selling price matrix contains the individual prices assigned to each product type.
• Multiplying these matrices yields a matrix presenting total potential sales, with calculations offering insights into expected earnings per product type.

Establishing selling prices allows organizations to forecast income and prepare financially for future operational needs.

Profit Calculation

Profit calculation involves determining the difference between what a business earns and spends. One efficient method is matrix operations, which produce insights into profits per product type over specific periods. In this exercise, taking the difference between the selling price matrix and the production cost matrix, combined with production data, allows for profit determination.

Consider this process:

• Subtract the production cost matrix from the selling price matrix to deduce potential profit per unit.
• Apply matrix multiplication with production data to calculate total profit for each product type.
• This operation yields a matrix that concisely expresses profits, aiding in strategic decision-making.

Such analysis is pivotal for companies like Acrosonic to assess profitability and optimize manufacturing strategies.