Problem 11 - 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 \(C-D\).

Step-by-Step Solution

Answer

The short answer for computing $C-D$ is: \(C - D = \left[\begin{array}{rrr}1 & 1 & -4\\-1 & -8 & 1\\6 & 3 & 1\end{array}\right]\).

Step 1: Identify the corresponding elements

Since both matrices C and D have the same dimensions (3 rows and 3 columns), we can easily identify the corresponding elements that need to be subtracted. For C: \(c_{11} = 3, c_{12} = -1, c_{13} = 0\) \(c_{21} = 2, c_{22} = -2, c_{23} = 3\) \(c_{31} = 4, c_{32} = 6, c_{33} = 2\) For D: \(d_{11} = 2, d_{12} = -2, d_{13} = 4\) \(d_{21} = 3, d_{22} = 6, d_{23} = 2\) \(d_{31} = -2, d_{32} = 3, d_{33} = 1\)

Step 2: Subtract corresponding elements

Now, subtract the corresponding elements from the two matrices. \(C - D = \left[\begin{array}{rrr} c_{11} - d_{11} & c_{12} - d_{12} & c_{13} - d_{13}\\ c_{21} - d_{21} & c_{22} - d_{22} & c_{23} - d_{23}\\ c_{31} - d_{31} & c_{32} - d_{32} & c_{33} - d_{33} \end{array}\right]\)

Step 3: Calculate the resulting matrix

Now perform the subtraction for each element and write down the resulting matrix. \[ C - D = \left[\begin{array}{rrr} 3 - 2 & -1 - (-2) & 0 - 4\\ 2 - 3 & -2 - 6 & 3 - 2\\ 4 - (-2) & 6 - 3 & 2 - 1 \end{array}\right] \] \[ C - D = \left[\begin{array}{rrr} 1 & 1 & -4\\ -1 & -8 & 1\\ 6 & 3 & 1 \end{array}\right] \] So, \(C - D = \left[\begin{array}{rrr}1 & 1 & -4\\-1 & -8 & 1\\6 & 3 & 1\end{array}\right]\)

Key Concepts

Linear Algebra

Linear Algebra is a fascinating branch of mathematics that focuses on vector spaces and the linear mappings between these spaces. It deals extensively with matrices and the various operations we can perform on them. Understanding linear algebra is crucial as it forms the backbone for numerous fields such as quantum mechanics, computer graphics, and data science.
One vital aspect is the study of matrices, which are essentially grids of numbers representing linear transformations. These matrices can be used to solve systems of linear equations, ultimately helping in modeling various real-world situations.
In linear algebra, operations like matrix addition, subtraction, and multiplication allow us to manipulate data in a structured manner. Each operation follows specific rules, maintaining the relationship and consistency between the datasets represented by these matrices.
Linear Algebra's tools provide a structured way to approach otherwise complex problems, offering generalized solutions that apply to more challenging multi-dimensional spaces.

Matrix Operations

Matrix operations are a set of algebraic manipulations that we can perform on matrices. These include addition, subtraction, and multiplication, among others. Each operation has rules that must be followed to maintain matrix symmetry and integrity.
When subtracting matrices, as shown in our exercise, it’s necessary that both matrices be of the same dimension. Subtraction is done by comparing and subtracting corresponding elements in these matrices.
For example, if we have two matrices, C and D, of the same size, then we subtract by:

• Identifying corresponding elements, e.g., element 1,1 of matrix C from element 1,1 of matrix D.
• Performing subtraction on these elements until all corresponding elements are processed.

The resulting matrix, C-D, can be viewed as a new matrix formed by these element-wise subtractions.
Matrix operations are fundamental in simplifying and understanding relationships between arrays of data. Mastery of matrix operations is the gateway to solving complex linear problems effectively.

Elementary Row Operations

Elementary Row Operations are actions that we can perform on the rows of a matrix to transform it into a different form, which is especially helpful in solving systems of equations. These operations include row swapping, row scaling (multiplying a row by a non-zero scalar), and row addition (adding or subtracting multiple of rows from another row).
These operations are pivotal in processes such as Gaussian elimination, where the goal is to simplify a matrix to either row-echelon form or reduced row-echelon form. While the primary aim is usually to solve a system of equations, these operations also aid in understanding matrix characteristics and performing decomposition.
It's essential to note that elementary row operations do not change the solution set of the associated system of linear equations. Therefore, these operations can be trusted to manipulate matrices without altering the inherent properties or solutions.
As we delve deeper into linear algebra, these operations provide a systematic method for tackling matrices, paving the way for algorithmic solutions in both hand computations and computer programming.