Problem 78 - Precalculus

Question

78\. Job Application. A company has two rubrics for scoring job applicants based on weighting education, experience, and the interview differently. Matrix \(A\) $$\left.\begin{array}{rrr} & \text { Rubric 1 } & \text { Rubric 2 } \\ \text { Education } & 0.5 & 0.6 \\ \text { Experience } & 0.3 & 0.1 \\ \text { Interview } & 0.2 & 0.3\end{array}\right]$$ Applicants reccive a score from 1 to 10 in cach catcgory (education, experience, and interview). Two applicants are shown in matrix \(B\) $$\begin{array}{lccc} & \text { Education } & \text { Experience } & \text { Interview } \\ \text { Applicant 1 } & 8 & 7 & 5 \\\\\text { Applicant 2 } & 6 & 8 & 8\end{array}$$ What is the order of \(B A ?\) What does each entry in \(B A\) tell us?

Step-by-Step Solution

Answer

Matrix \( BA \) is \( 2 \times 2 \). Each entry represents an applicant's score under a rubric.

Step 1: Understand Matrices A and B

Matrix \( A \) contains the weights for scoring in two rubrics. Each column represents a rubric, with the rows representing the weights for Education, Experience, and Interview respectively. Matrix \( B \) contains the scores for two applicants in these categories. Each row corresponds to an applicant, and each column corresponds to a category.

Step 2: Determine Matrix Dimensions for Multiplication

Matrix \( B \) is a \( 2 \times 3 \) matrix (2 applicants, 3 categories), and Matrix \( A \) is a \( 3 \times 2 \) matrix (3 categories, 2 rubrics). When multiplying a \( 2 \times 3 \) matrix by a \( 3 \times 2 \) matrix, the resulting matrix will be \( 2 \times 2 \), representing the scores for each applicant under each rubric.

Step 3: Multiply Matrices A and B

Multiply each row of Matrix \( B \) by each column of Matrix \( A \). Calculate the score for Applicant 1 under Rubric 1 as \((8 \times 0.5) + (7 \times 0.3) + (5 \times 0.2)\). Do the same for Rubric 2, then apply the same process for Applicant 2.

Step 4: Calculate the Scores

For Applicant 1, Rubric 1: \[ (8 \times 0.5) + (7 \times 0.3) + (5 \times 0.2) = 4 + 2.1 + 1 = 7.1 \]Rubric 2:\[ (8 \times 0.6) + (7 \times 0.1) + (5 \times 0.3) = 4.8 + 0.7 + 1.5 = 7.0 \]For Applicant 2, Rubric 1:\[ (6 \times 0.5) + (8 \times 0.3) + (8 \times 0.2) = 3 + 2.4 + 1.6 = 7.0 \]Rubric 2:\[ (6 \times 0.6) + (8 \times 0.1) + (8 \times 0.3) = 3.6 + 0.8 + 2.4 = 6.8 \]

Step 5: Interpret the Matrix BA

The resulting matrix \( BA \) is \( \begin{bmatrix} 7.1 & 7.0 \ 7.0 & 6.8 \end{bmatrix} \). Each row represents an applicant, and each column the score under Rubric 1 and Rubric 2. For example, the first row (7.1, 7.0) signifies that Applicant 1 scores 7.1 using Rubric 1 and 7.0 using Rubric 2.

Key Concepts

Matrix Order

In mathematics, whenever we work with matrices, understanding their order is crucial. The order of a matrix is given by the number of rows and columns it contains. It's described in the format "rows x columns." For the problem at hand, Matrix \( B \) is a \( 2 \times 3 \) matrix, meaning it has 2 rows and 3 columns. Matrix \( A \) is a \( 3 \times 2 \) matrix, with 3 rows and 2 columns. To multiply these two matrices, it is important to pay attention to the inside dimensions (3 and 3 in our case), which determine if multiplication is possible. If these numbers match, the multiplication can proceed, and the resulting matrix is formed by the outside dimensions (2 and 2). Therefore, the product \( BA \) results in a \( 2 \times 2 \) matrix, which is perfect for our application scoring system example.

Matrix Dimensions

Matrix dimensions are a fundamental aspect of matrix theory, and they determine many properties of the matrix, including how and if matrices can be operated on, like through addition or multiplication. Each matrix has dimensions defined by its number of rows and columns. A matrix \( M \) that has \( r \) rows and \( c \) columns is expressed as an \( r \times c \) matrix. The example problem shows Matrix \( B \) as a \( 2 \times 3 \) matrix reflecting two applicants being scored across three different categories: education, experience, and interview. On the other hand, Matrix \( A \) is a \( 3 \times 2 \) matrix, representing the weights given to each category under two different scoring rubrics. When considering matrix multiplication, the number of columns in the first matrix must equal the number of rows in the second matrix. Here, Matrix \( B's \) 3 columns align with Matrix \( A's \) 3 rows, allowing the multiplication to proceed smoothly.

Scoring Systems

In the context of this exercise, scoring systems are devised using matrices to systematically evaluate job applicants. Each applicant's raw scores across several categories like education, experience, and the interview are represented in Matrix \( B \). These are then adjusted or weighted using a set of rubrics found in Matrix \( A \). Matrix \( A \) contains different weights applied to these categories to form distinct scoring approaches (Rubric 1 and Rubric 2, in this case). When the matrices are multiplied, we obtain Matrix \( BA \), which provides a final score for each applicant based on each rubric. This method allows for a clear, quantitative comparison of applicants according to diverse hiring criteria. This system, by varying weights in the matrix, can prioritize certain categories over others, thus customizing the scoring outcomes to suit the company's needs. This flexibility demonstrates the power and utility of matrix multiplication in real-world decision-making scenarios.