Problem 13 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Perform the indicated operations. \(\left[\begin{array}{lll}6 & 3 & 8 \\ 4 & 5 & 6\end{array}\right]-\left[\begin{array}{lll}3 & -2 & -1 \\ 0 & -5 & -7\end{array}\right]\)

Step-by-Step Solution

Answer

The result of the subtraction is: \(\begin{bmatrix} 6 & 3 & 8 \\ 4 & 5 & 6 \end{bmatrix} - \begin{bmatrix} 3 & -2 & -1 \\ 0 & -5 & -7 \end{bmatrix} = \begin{bmatrix} 3 & 5 & 9 \\ 4 & 10 & 13 \end{bmatrix}\)

Step 1: Verify the dimensions of the matrices

Make sure that both matrices have the same dimensions. Both of them are 2x3, so we can proceed to the subtraction. Matrix 1: \(\begin{bmatrix} 6 & 3 & 8 \\ 4 & 5 & 6 \end{bmatrix}\) Matrix 2: \(\begin{bmatrix} 3 & -2 & -1 \\ 0 & -5 & -7 \end{bmatrix}\)

Step 2: Subtract corresponding entries

Now, subtract the entries of the second matrix from the corresponding entries of the first matrix: \((6-3, 3-(-2), 8-(-1)),\\(4-0, 5-(-5), 6-(-7))\)

Step 3: Simplify the resulting matrix

Simplify the obtained matrix by performing the arithmetic operations: \(\begin{bmatrix} 3 & 5 & 9 \\ 4 & 10 & 13 \end{bmatrix}\) Therefore, the result of the subtraction is: \(\begin{bmatrix} 6 & 3 & 8 \\ 4 & 5 & 6 \end{bmatrix} - \begin{bmatrix} 3 & -2 & -1 \\ 0 & -5 & -7 \end{bmatrix} = \begin{bmatrix} 3 & 5 & 9 \\ 4 & 10 & 13 \end{bmatrix}\)

Key Concepts

Matrix Operations

Working with matrices involves various operations, such as addition, subtraction, and multiplication. Matrix subtraction is a fundamental process in this realm. Understanding matrix operations is essential for many fields, ranging from computer graphics to data analysis. When performing matrix subtraction, it is crucial to ensure that the matrices involved have the same dimensions, meaning they should have the same number of rows and columns.

Here's a simple breakdown of how matrix subtraction works:

• **Element-Wise Subtraction**: Each element in one matrix is paired with the corresponding element in the other matrix and the subtraction is performed element by element.
• **Maintain Dimensions**: The new matrix formed after subtraction will have the same dimensions as the original matrices.

This process results in a new matrix where each position holds the result of subtracting elements from the two matrices that correspond to that position.

Mastering this operation requires practice and it forms the basis for more complex matrix manipulations.

Elementary Algebra

Elementary algebra involves operations and manipulations of mathematical symbols and numbers to solve problems. When dealing with matrix subtraction, algebraic concepts come into play since you operate on numbers according to mathematical rules.

To understand this in the context of matrix subtraction, consider the step where you have to subtract individual matrix elements:

• The **Subtraction of Signed Numbers**: Keep in mind the rules for subtracting positives and negatives. For instance, subtracting a negative number is equivalent to performing an addition.
• **Simplification**: After performing the subtraction, simplify by combining terms where applicable. This step ensures the resulting matrix elements are reduced to their simplest form.

These elementary algebra principles simplify operations, helping in obtaining clear and correct results. They underscore the importance of a strong grasp of basic arithmetic and signed number operations.

Applied Mathematics

Applied mathematics takes theory and data to solve real-world problems. Matrix operations, including subtraction, play significant roles in areas like engineering, physics, and even economics.

One practical use of matrix subtraction is in computational tasks. For instance:

• **Image Processing**: Matrix subtraction can be used to detect changes between images, crucial in areas like security and medical imaging.
• **Finance**: It helps in analyzing data sets and in financial modeling where large data arrays are manipulated.

Understanding how to perform these operations effectively allows professionals to manipulate datasets, model scenarios, and derive insights that inform decision-making in various practical applications.

A clear grasp of applied mathematics empowers one to translate numerical data into actionable solutions across disciplines.