Problem 31 - A Graphical Approach to College Algebra

Question

Perform each operation if possible. $$3\left[\begin{array}{rrr} 6 & -1 & 4 \\ 2 & 8 & -3 \\ -4 & 5 & 6 \end{array}\right]+5\left[\begin{array}{rrr} -2 & -8 & -6 \\ 4 & 1 & 3 \\ 2 & -1 & 5 \end{array}\right]$$

Step-by-Step Solution

Answer

The result is \( \begin{bmatrix} 8 & -43 & -18 \\ 26 & 29 & 6 \\ -2 & 10 & 43 \end{bmatrix} \).

Step 1: Distribute the Scalar to the First Matrix

Multiply each element of the first matrix by the scalar 3.\[3 \times \begin{bmatrix} 6 & -1 & 4 \ 2 & 8 & -3 \ -4 & 5 & 6 \end{bmatrix} = \begin{bmatrix} 18 & -3 & 12 \ 6 & 24 & -9 \ -12 & 15 & 18 \end{bmatrix}\]

Step 2: Distribute the Scalar to the Second Matrix

Multiply each element of the second matrix by the scalar 5.\[5 \times \begin{bmatrix} -2 & -8 & -6 \ 4 & 1 & 3 \ 2 & -1 & 5 \end{bmatrix} = \begin{bmatrix} -10 & -40 & -30 \ 20 & 5 & 15 \ 10 & -5 & 25 \end{bmatrix}\]

Step 3: Add the Two Resulting Matrices

Add the corresponding elements of the resulting matrices from Steps 1 and 2.\[\begin{bmatrix} 18 & -3 & 12 \ 6 & 24 & -9 \ -12 & 15 & 18 \end{bmatrix} + \begin{bmatrix} -10 & -40 & -30 \ 20 & 5 & 15 \ 10 & -5 & 25 \end{bmatrix} = \begin{bmatrix} 8 & -43 & -18 \ 26 & 29 & 6 \ -2 & 10 & 43 \end{bmatrix}\]

Key Concepts

Scalar Multiplication

Scalar multiplication is an operation in linear algebra where every element of a matrix is multiplied by a single number, known as a scalar. This process changes the size of each number within the matrix, but the overall structure, like rows and columns, stays the same.
When you perform scalar multiplication, you take each element within the matrix and multiply it by the scalar. For example, if you have a scalar of 3 and a matrix like this: \\( \begin{bmatrix} 6 & -1 & 4 \ 2 & 8 & -3 \ -4 & 5 & 6 \end{bmatrix} \), you would multiply each element, such that 3 becomes 3 times 6, 3 times -1, etc.
Here’s a step-by-step guide to scalar multiplication:

• Identify the scalar and the matrix.
• Multiply each element of the matrix by the scalar.
• Ensure that the number of rows and columns remains unchanged.
• Write down the resultant matrix.

Scalar multiplication is fundamental because it’s typically the first step in more complex operations like matrix transformations and scaling in various applications.

Matrix Addition

Matrix addition is another operation in linear algebra that involves adding together two matrices by summing their corresponding elements. For matrix addition to work, both matrices must be of the same size, meaning they have the same number of rows and columns.
In the exercise, after performing scalar multiplication on two separate matrices, we next perform matrix addition. We added the matrices: \\( \begin{bmatrix} 18 & -3 & 12 \ 6 & 24 & -9 \ -12 & 15 & 18 \end{bmatrix} \) with \\( \begin{bmatrix} -10 & -40 & -30 \ 20 & 5 & 15 \ 10 & -5 & 25 \end{bmatrix} \) by adding each corresponding element.
Here are the steps to follow for matrix addition:

• Ensure matrices have the same dimensions.
• Add the elements at the same position in each matrix.
• Place the result in the corresponding position in a new matrix.

Matrix addition allows you to combine multiple matrices, which is crucial in fields that deal with multiple variable systems and require data aggregation.

Linear Algebra

Linear algebra is a branch of mathematics that focuses on vectors, vector spaces, and the linear transformations and operations on these structures. It plays a critical role in engineering, physics, computer science, and statistics, providing tools for solving systems of linear equations, among other applications.
One core aspect of linear algebra is handling matrices, which are rectangular arrays of numbers that represent complex data. The operations, such as scalar multiplication and matrix addition, are tools in linear algebra that help simplify and solve complex problems involving matrices and vectors.
Applications of linear algebra include:

• Computer graphics, where matrices are used for rotations and transformations.
• Data science, to handle and manipulate large datasets efficiently.
• Machine learning, particularly in algorithms involving dimensionality reduction.

Understanding linear algebra is foundational for anyone working in quantitative fields, emphasizing logical reasoning and problem-solving abilities.

Matrices

Matrices are a fundamental concept in linear algebra. They are collections of numbers arranged in rows and columns forming a rectangular shape. Each number in the matrix is called an element, and matrices serve as a compact way to represent linear equations or transformations.
In matrix notation, rows run horizontally from left to right, and columns run vertically from top to bottom. For instance, the exercise involves two 3x3 matrices: \\( \begin{bmatrix} 6 & -1 & 4 \ 2 & 8 & -3 \ -4 & 5 & 6 \end{bmatrix} \) and \\( \begin{bmatrix} -2 & -8 & -6 \ 4 & 1 & 3 \ 2 & -1 & 5 \end{bmatrix} \). Both have three rows and three columns, hence they are said to be of the same order.
Properties of matrices include:

• Accurate structure: each row and column must contain elements.
• Eligible for various operations like addition, multiplication, and transformations.
• Used to model real-world systems mathematically.

Matrices are extremely powerful tools and appear in many contexts, such as solving systems of equations, representing networks, and more. Mastering them unlocks many doors in the study of advanced mathematics and its applications.