Problem 9 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Refer to the following matrices: \(A=\left[\begin{array}{rr}-1 & 2 \\ 3 & -2 \\ 4 & 0\end{array}\right] \quad B=\left[\begin{array}{rr}2 & 4 \\ 3 & 1 \\ -2 & 2\end{array}\right]\) \(C=\left[\begin{array}{rrr}3 & -1 & 0 \\ 2 & -2 & 3 \\ 4 & 6 & 2\end{array}\right] \quad D=\left[\begin{array}{rrr}2 & -2 & 4 \\ 3 & 6 & 2 \\\ -2 & 3 & 1\end{array}\right]\) Compute \(A+B\).

Step-by-Step Solution

Answer

The sum of matrices A and B is: \(A+B = \left[\begin{array}{rr} 1 & 6 \\\ 6 & -1 \\\ 2 & 2\end{array}\right]\).

Step 1: Verify that the matrices can be added

Both matrices A and B have dimensions 3x2, which means they both have 3 rows and 2 columns. Since they have the same dimensions, they can be added together.

Step 2: Set up an empty 3x2 matrix for the sum

We will create an empty 3x2 matrix for the sum of matrices A and B, which we will fill with the results of our addition in the next step: \(A+B = \left[\begin{array}{rr} & \\\ & \\\ & \end{array}\right]\).

Step 3: Add corresponding elements of matrices A and B

Now, we add the elements in the same positions in matrices A and B to find the sum: \((A+B)_{11} = A_{11} + B_{11} = -1 + 2 = 1\) \((A+B)_{12} = A_{12} + B_{12} = 2 + 4 = 6\) \((A+B)_{21} = A_{21} + B_{21} = 3 + 3 = 6\) \((A+B)_{22} = A_{22} + B_{22} = -2 + 1 = -1\) \((A+B)_{31} = A_{31} + B_{31} = 4 + (-2) = 2\) \((A+B)_{32} = A_{32} + B_{32} = 0 + 2 = 2\)

Step 4: Write the sum matrix

Finally, we fill the results into the sum matrix: \(A+B = \left[\begin{array}{rr} 1 & 6 \\\ 6 & -1 \\\ 2 & 2\end{array}\right]\) So, the sum of matrices A and B is: \(A+B = \left[\begin{array}{rr} 1 & 6 \\\ 6 & -1 \\\ 2 & 2\end{array}\right]\).

Key Concepts

Matrix Operations

Matrix operations form a core part of linear algebra and are crucial for many mathematical and real-world applications. One of the primary operations in working with matrices is matrix addition. Just like adding numbers, matrix addition involves combining the elements. However, it's important to ensure that the matrices being added are compatible; they must have the same dimensions. If matrix A is 3 rows by 2 columns, matrix B must also be 3 rows by 2 columns for the addition to be possible.

The process of adding matrices is straightforward:

• Identify matrices with identical dimensions.
• Set up an empty matrix with the same dimensions to store the results.
• Add corresponding elements from both matrices to get the resulting matrix.

This operation maintains the dimensionality of the matrices, meaning that the resulting sum will have the same dimensions as the original matrices. Understanding these operations is fundamental when dealing with matrices in mathematics as they prepare the groundwork for more complex calculations.

Linear Algebra Applications

Linear algebra is more than just abstract theory—it has powerful applications in computer graphics, engineering, physics, and beyond. Matrix operations like addition are used extensively in these fields to perform calculations that handle complex data efficiently. For instance, in computer graphics, matrices are pivotal in transformations, rotations, and translations of images.

Matrix addition can also model various real-life problems. In economics, matrices can represent different equations modeling key economic factors, allowing analysts to predict outcomes through solution matrices. In robotics and AI, matrices help in modeling environments and decision matrices, thus aiding in developing sophisticated algorithms.

These applications demonstrate how essential matrix calculations are not only in theoretical mathematics but also in practical, impactful fields. Mastery of matrix operations like addition is a stepping stone to understanding and leveraging the broader potential of linear algebra in innovative solutions.

Matrices in Mathematics

Matrices play a crucial role in both elementary and advanced aspects of mathematics. They serve as the building blocks for systems of linear equations and have a wide range of applications throughout the mathematical disciplines.

In mathematics, matrices can represent systems of linear equations compactly, making large data sets more manageable. Solving these systems using matrix operations can result in finding values that satisfy multiple equations simultaneously.

Additionally, matrices facilitate operations like determining eigenvalues and eigenvectors, which are essential in understanding matrix behavior in transformations. They're also vital in defining and working with linear transformations, which are used comprehensively in finite-dimensional vector spaces.

Understanding matrices in mathematics opens a pathway to exploring more complex mathematical concepts and provides the necessary tools for application in real-world scenarios, where mathematical modeling is used for predictions and optimizations.