Problem 19 - Algebra 1

Question

Find the difference of the matrices. $$ \left[\begin{array}{rr} -4 & 1 \\ 0 & -13 \\ 2 & -8 \end{array}\right]-\left[\begin{array}{rr} -6 & 3 \\ -5 & 8 \\ 2 & -7 \end{array}\right] $$

Step-by-Step Solution

Answer

The difference between the two given matrices is:\[\left[\begin{array}{rr} 2 & -2 \ 5 & -21 \ 0 & -1\end{array}\right]\]

Step 1: Identify Corresponding Elements

First, identify the elements that are in the same position in both matrices for subtraction. The matrices have the same sizes, so all corresponding element positions in the two matrices are the same.

Step 2: Subtract Corresponding Elements

Next, Subtract each element in the second matrix from the corresponding element in the first matrix. This will give your difference matrix: \[\left[\begin{array}{rr} -4-(-6) & 1-3 \ 0-(-5) & -13-8 \ 2-2 & -8-(-7)\end{array}\right]\]

Step 3: Calculation of Each Element

Finally, perform the subtraction for each element. This will give the final difference between the two matrices: \[\left[\begin{array}{rr} 2 & -2 \ 5 & -21 \ 0 & -1\end{array}\right]\]

Key Concepts

Matrices

A matrix is a rectangular array of numbers arranged in rows and columns. It looks like a grid filled with numbers, and each number in the matrix is called an element. To define a matrix, we denote its size by its number of rows and columns, commonly referred to as its dimensions. For instance, if a matrix has 3 rows and 2 columns, it is a \(3 \times 2\) matrix.

Matrices are used in various fields such as mathematics, engineering, physics, and computer science. They provide a way to structure data and perform calculations systematically. You can think of matrices like spreadsheets where numbers are organized in a neat manner, allowing you to perform complex operations easily.

Understanding matrices is essential because they form the foundation for matrix operations like addition, subtraction, multiplication, and more. These operations are crucial in solving systems of equations, transforming geometric shapes, and processing digital data in applications like computer graphics.

Matrix Operations

Matrix operations are a set of rules that allow us to manipulate matrices. Some common operations include addition, subtraction, and multiplication. Each operation follows specific rules and requires matrices to meet particular conditions to be performed.

For subtraction, it is essential that the matrices involved have the same dimensions. This means they must have the same number of rows and columns. Only when both matrices are of the same size, can we perform element-by-element subtraction. The result is another matrix of the same dimensions.

Other matrix operations also include:

• Matrix Addition: Similar to subtraction, but instead of subtracting, we add corresponding elements.
• Matrix Multiplication: More complex, as it involves the sum of products of rows and columns and requires specific conditions for dimensions.

Matrix subtraction, along with these operations, is foundational for more advanced concepts, such as solving linear equations or performing transformations in graphics.

Element Subtraction

Element subtraction is a straightforward process where we subtract corresponding elements of two matrices. To perform element subtraction:

• Identify the corresponding elements in each matrix. In matrix subtraction, each element in the first matrix is subtracted by the element in the same position in the second matrix.
• Perform the subtraction: for every element in the matrices, subtract the second matrix element from the first matrix element.
• The result is a new matrix, known as the difference matrix, that contains the outcome of each element subtraction.

This process often involves negative numbers, so be cautious with signs. For instance, subtracting a negative number is equivalent to adding the positive version of that number. In this way, element subtraction fosters understanding of not only matrix operations but also basic arithmetic with positive and negative numbers.

Element subtraction in matrices serves as an excellent exercise in systematic calculation, helping to develop both an understanding of matrix operations and arithmetic skills.