Problem 17 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Perform the indicated operations. \(\left[\begin{array}{lll}1.2 & 4.5 & -4.2 \\ 8.2 & 6.3 & -3.2\end{array}\right]-\left[\begin{array}{rrr}3.1 & 1.5 & -3.6 \\ 2.2 & -3.3 & -4.4\end{array}\right]\)

Step-by-Step Solution

Answer

The short answer to the matrix subtraction is: \(\left[\begin{array}{lll}-1.9 & 3 & -0.6 \\\ 6 & 9.6 & 1.2\end{array}\right]\)

Step 1: Identify the corresponding elements in each matrix

In this step, we will identify the elements in the same position from each matrix that needs to be subtracted. Matrix A: \(\left[\begin{array}{lll}1.2 & 4.5 & -4.2 \\\ 8.2 & 6.3 & -3.2\end{array}\right]\) Matrix B: \(\left[\begin{array}{rrr}3.1 & 1.5 & -3.6 \\\ 2.2 & -3.3 & -4.4\end{array}\right]\)

Step 2: Subtract corresponding elements

Here, subtract the numbers that were identified in the previous step and fill the result matrix accordingly, i.e., (A11 - B11), (A12 - B12), and so on. Result Matrix: \(\left[\begin{array}{lll}(1.2-3.1) & (4.5-1.5) & (-4.2+3.6) \\\ (8.2-2.2) & (6.3+3.3) & (-3.2+4.4)\end{array}\right]\)

Step 3: Compute the subtraction

Now, perform the subtraction for each element in the result matrix. \(\left[\begin{array}{lll}-1.9 & 3 & -0.6 \\\ 6 & 9.6 & 1.2\end{array}\right]\) So, the final result after subtracting the two matrices is: \(\left[\begin{array}{lll}-1.9 & 3 & -0.6 \\\ 6 & 9.6 & 1.2\end{array}\right]\)

Key Concepts

Matrix Operations

Matrix operations involve performing arithmetic on matrices, which are structures made up of rows and columns filled with elements, typically numbers. One fundamental matrix operation is matrix subtraction, which involves subtracting one matrix from another, element by element.
For the subtraction to be valid, both matrices must have the same dimensions, meaning they have the same number of rows and columns.

• Step-by-step Subtraction: To subtract two matrices, identify the corresponding elements, or positions, in each matrix and perform the subtraction for each pair of elements.
• Element-wise Subtraction: The result is a new matrix where each element is the difference between corresponding elements in the original matrices.

In this specific exercise, we performed element-wise subtraction of two matrices of size 2x3, resulting in a new matrix of the same size.

Elementary Arithmetic

Elementary arithmetic is the foundation of all mathematical operations and includes basic operations such as addition, subtraction, multiplication, and division. In matrix subtraction, we specifically use the subtraction operation.
For matrix elements,

• Subtraction: Take two numbers, say "a" and "b", and perform "a - b" to get the resulting value for the position in the result matrix.
• Handling Signs: Pay extra attention to the signs of numbers, especially when subtracting negative numbers, as they can turn into additional operations (e.g., subtracting a negative becomes addition).

This exercise showed careful application of subtraction by explicitly subtracting corresponding elements from each matrix, resulting in our final matrix answer.

Linear Algebra

Linear algebra extends the arithmetic operations to matrix operations and is a branch of mathematics concerning linear equations, vectors, and matrices. Understanding matrix subtraction is crucial because it applies to more complex problems in linear algebra, such as solving systems of equations and transformations.

• Matrix as Linear Map: A matrix can be thought of as a linear transformation when acting on vectors. Subtraction modifies these transformations.
• Applications of Subtraction: Recognizing that matrix operations, like subtraction, translate to real-world applications in computer graphics, statistics, and scientific computation.

The exercise performed is a simple illustration within linear algebra of how matrices are manipulated and form the basis for solving more complex equations and transformations encountered in the field.