Problem 15 - Discrete Mathematics with Applications

Question

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

Step-by-Step Solution

Answer

The result of the matrix subtraction \(B - C\) is: \[B - C = \left[\begin{array}{rrr}{3} & {-2} & {5} \\\ {0} & {-1} & {-1}\end{array}\right]\]

Step 1: Display the matrices B and C.

Write down the given matrices. Matrix B: \[B = \left[\begin{array}{rrr}{0} & {-2} & {5} \\\ {0} & {0} & {1}\end{array}\right]\] Matrix C: \[C = \left[\begin{array}{rrr}{-3} & {0} & {0} \\\ {0} & {1} & {2}\end{array}\right]\]

Step 2: Verify if the matrices have the same dimensions.

In order to perform matrix subtraction, both matrices must have the same dimensions. In this case, each matrix has 2 rows and 3 columns, so they have the same dimensions, and we are able to subtract them.

Step 3: Subtract Matrices B and C.

To subtract matrices, we subtract corresponding elements in the given matrices. So the difference B - C is a matrix D with elements: \[d_{i,j} = b_{i,j} - c_{i,j}\] where \(d_{i,j}\) represents the element in the i-th row and j-th column of matrix D, and \(b_{i,j}\) and \(c_{i,j}\) are the corresponding elements in matrices B and C, respectively. Now, let's calculate the elements of matrix D: 1st row, 1st column: \(0 - (-3) = 3\) 1st row, 2nd column: \(-2 - 0 = -2\) 1st row, 3rd column: \(5 - 0 = 5\) 2nd row, 1st column: \(0 - 0 = 0\) 2nd row, 2nd column: \(0 - 1 = -1\) 2nd row, 3rd column: \(1 - 2 = -1\) So, the resulting matrix D is: \[D = \left[\begin{array}{rrr}{3} & {-2} & {5} \\\ {0} & {-1} & {-1}\end{array}\right]\]

Step 4: Write down the answer.

Now we have found the result of the matrix subtraction "B - C", which is matrix D: \[B - C = \left[\begin{array}{rrr}{3} & {-2} & {5} \\\ {0} & {-1} & {-1}\end{array}\right]\]

Key Concepts

element-wise subtraction

Matrix subtraction, which involves taking one matrix from another, relies on a critical operation known as element-wise subtraction. Think of this as subtracting each element of one matrix from the corresponding element of another. For instance, in the case of matrices B and C, this means calculating the difference between each matched pair of numbers across the entire matrix. By following this method, we are aiming to create a new matrix where each position contains the result of these individual subtractions.
Understanding element-wise subtraction helps in visualizing actions on matrices in a simple manner. Here’s a quick guide:

• You begin with two matrices of the same dimensions.
• Subtract the corresponding elements of the two matrices.
• Record the results in a new matrix with the same dimensions.

Through practice, this process becomes intuitive and enhances your ability to perform other matrix operations.

matrix dimensions

For matrix operations like subtraction to work, it's crucial that the matrices involved have the same dimensions. Matrix dimensions are denoted as ‘rows by columns’. In our example, both matrices B and C are 2x3 matrices, meaning they both have 2 rows and 3 columns.
This similarity in dimension ensures that every element in one matrix has a corresponding element in the other, making it possible to carry out element-wise operations. Checking dimensions is often the first step in any matrix operation, because mismatched matrices can't be processed together with these operations. Here's why dimensions matter:

• Element-wise operations require each element to have a match in the opposite matrix.
• Consistency in dimensions ensures mathematical operations can proceed correctly.
• It simplifies anticipating the shape of the resulting matrix from the operation.

Practically, before performing any matrix subtraction, always inspect the matrices to confirm they align in dimensions.

2D arrays

Matrices, which can be visualized as 2D arrays, form a core part of understanding operations like subtraction between matrix pairs. Conceptually, you can think of a matrix as a grid-like structure where numbers are organized into rows and columns—a classic representation of a 2D array.
This structure allows us to perform various operations systematically. In computer science and mathematics, managing data in 2D arrays simplifies complex calculations and data organization. Understanding these foundational elements is a huge advantage:

• They provide an orderly way to handle sets of numbers systematically.
• Operations like addition and subtraction are performed using a predictable pattern.
• Visualization is easier when data is organized in an array format.

Embracing the matrix as a 2D array opens up a world of efficient computation and data handling possibilities, laying the groundwork for advanced operations and problem-solving.