Problem 51 - College Algebra and Calculus: An Applied Approach

Question

A corporation that makes sunglasses has four factories, each of which manufactures two products. The number of units of product \(i\) produced at factory \(j\) in one day is represented by \(a_{i j}\) in the matrix \(A=\left[\begin{array}{rrrr}100 & 120 & 60 & 40 \\ 140 & 160 & 200 & 80\end{array}\right] .\) Find the production levels if production is increased by \(10 \%\). (Hint: Because an increase of \(10 \%\) corresponds to \(100 \%+10 \%\), multiply the matrix by \(1.10 .\))

Step-by-Step Solution

Answer

After multiplying each entry in the matrix 'A' by 1.10 as per the scalar multiplication rule, the resulting matrix 'B', representing a 10% increase in daily production at each factory for both products, would be \[B=\left[\begin{array}{rrrr}110 & 132 & 66 & 44 \ 154 & 176 & 220 & 88\end{array}\right].\]

Step 1: Understanding the given matrix and scalar

We are given a 2x4 matrix 'A', which represents the daily production of two products in four factories. A scalar of 1.10 is provided, which represents a 10% increase in production levels.

Step 2: Multiplying the matrix by the scalar

To calculate the new production levels, each entry in the matrix 'A' should be multiplied by the scalar 1.10 using the matrix scalar multiplication rule. This rule states that to multiply a matrix by a scalar, every entry in the matrix is multiplied by the scalar. In this case, every daily production figure 'a_{ij}' in the matrix should be multiplied by 1.10.

Step 3: Calculating the resulting matrix

After performing the multiplication, the resulting matrix 'B' represents the new daily production levels of each product in each factory. The entries of 'B' are calculated as follows: \[b_{ij} = 1.10 * a_{ij}\] Apply this calculation for each entry in the matrix.

Step 4: Interpretation of the resulting matrix

The resulting 2x4 matrix 'B' represents the new daily production levels for each product in each factory after a 10% increase in production. Each entry 'b_{ij}' in the matrix 'B' denotes the new daily production level of product 'i' at factory 'j'.

Key Concepts

Matrix Operations

Matrix operations are mathematical procedures that involve matrices, which are rectangular arrays of numbers arranged in rows and columns. One fundamental operation is matrix scalar multiplication, where each element of a matrix is multiplied by a scalar, which is a single number. This operation adjusts every value in the matrix by the scalar amount. For example, if we have a matrix \(A\) and we multiply it by a scalar \(k\), the outcome is a new matrix where each element \(a_{ij}\) in \(A\) becomes \(b_{ij} = k \, a_{ij}\). This operation is useful in various practical applications, such as adjusting production outputs or financial figures by a percentage, as seen in the sunglasses factory example where each production number is increased by 10% by multiplying by 1.10.

Production Increase Calculation

Calculating an increase in production levels using matrix operations is straightforward with scalar multiplication. In the example of the sunglasses corporation, each factory increases production by 10%. This can be represented by a scalar of 1.10.

• The original production numbers are organized in a 2x4 matrix, representing two products across four factories.
• To account for the 10% increase, each number in this matrix is multiplied by 1.10.

This results in a new set of production numbers, where each entry is 10% higher than the original. The multiplication is applied uniformly to every element, ensuring consistent increases across all values. This method allows for efficient and accurate updates to production figures in scenarios where uniform percentage changes are required.

Algebraic Applications

Matrix operations, including scalar multiplication, have wide applications in algebra and various fields. In business, they facilitate the modeling of changes in production, costs, or sales. By scaling matrices with a scalar, we can simulate real-world percentage changes seamlessly. This method is not only efficient but also scalable. For instance, if a corporation wants to predict future production scenarios, multiplying current matrix data by forecasted growth rates provides immediate insights. In algebraic contexts, matrices simplify dealing with multiple variables at once. This is particularly useful in large-scale computations, where traditional arithmetic would be cumbersome. By using matrix algebra, businesses, scientists, and engineers can solve complex systems more effectively, making it a crucial tool in decision-making processes.