Problem 6 - Discrete Mathematics with Applications

Question

Let \(A=\left[\begin{array}{ccc}1 & 0 & -1 \\ 0 & 2 & 3\end{array}\right], B=\left[\begin{array}{ccc}0 & -2 & 5 \\ 0 & 0 & 1\end{array}\right],\) and \(C=\left[\begin{array}{ccc}-3 & 0 & 0 \\ 0 & 1 & 2\end{array}\right] .\) Find each. $$A-B$$

Step-by-Step Solution

Answer

The result of the matrix subtraction \(A - B\) is given by: \[A - B = \begin{bmatrix} 1 & 2 & -6 \\ 0 & 2 & 2 \\ \end{bmatrix}\]

Step 1: Identify the size of the matrices

Both matrices A and B have dimensions 2x3, which means they have 2 rows and 3 columns.

Step 2: Subtract the corresponding elements

To perform A - B, we need to subtract the corresponding elements of both matrices. This means subtracting each element of matrix B from the corresponding element in matrix A: \[\begin{bmatrix} (1 - 0) & (0 - (-2)) & (-1 - 5) \\ (0 - 0) & (2 - 0) & (3 - 1) \\ \end{bmatrix}\]

Step 3: Calculate the result

Now, we calculate the result of the subtraction for each element in the matrices: \[\begin{bmatrix} 1 & 2 & -6\\ 0 & 2 & 2\\ \end{bmatrix}\] So, A - B is given by the matrix: \[A - B = \begin{bmatrix} 1 & 2 & -6\\ 0 & 2 & 2\\ \end{bmatrix}\]

Key Concepts

Matrix Operations

Matrix operations involve various techniques for manipulating matrices, which are collections of numbers arranged in rows and columns. They're a crucial part of linear algebra and are used in different fields such as computer graphics, physics, and statistics. Some common matrix operations include matrix addition, subtraction, multiplication, and finding determinants. These operations follow specific rules and often require matrices to meet certain conditions, such as having the same dimensions for addition or subtraction. In this exercise, we focus on matrix subtraction, specifically subtracting matrix B from matrix A. Ensuring that these matrices have compatible dimensions is vital.

Element-Wise Subtraction

Element-wise subtraction is a fundamental matrix operation. In matrix subtraction, the elements of one matrix are subtracted from corresponding elements of another matrix. For example, with matrices A and B given in the exercise, element-wise subtraction means:

• Subtracting each element of matrix B from the same position in matrix A

Let's break this down with the exercise:

• The first element in the first row and column of A is 1, and in B is 0. So, we subtract 0 from 1 to get 1.
• Similarly, the second element in the first row is 0 in A and -2 in B. Subtracting gives us 0 - (-2) = 2.
• We continue this process for all elements, resulting in the matrix: \[ \begin{bmatrix}1 & 2 & -6\0 & 2 & 2\\end{bmatrix}\]

Element-wise subtraction only works when both matrices have the same dimensions, ensuring that each position in one matrix corresponds directly to a position in the other.

Matrix Dimensions

Understanding matrix dimensions is essential as it determines the feasibility of performing operations like addition or subtraction. A matrix’s dimension is denoted as 'm x n', where 'm' is the number of rows, and 'n' is the number of columns. In our example, matrices A and B both have dimensions of 2x3:

• 2 rows
• 3 columns

For the subtraction operation \(A - B\) to be valid, both matrices must possess the same dimensions. This alignment allows each element in a row and a column of A to line up perfectly with the corresponding element in B. If the dimensions mismatch, such operations are impossible. In advanced matrix operations, understanding and verifying matrix dimensions are the first and crucial steps.