Question

If \(\mathbf{A}=\left(\begin{array}{rrr}{3} & {2} & {-1} \\ {2} & {-1} & {2} \\\ {1} & {2} & {1}\end{array}\right)\) and \(\mathbf{B}=\left(\begin{array}{rrr}{2} & {1} & {-1} \\ {-2} & {3} & {3} \\\ {1} & {0} & {2}\end{array}\right),\) verify that \(2(\mathbf{A}+\mathbf{B})=2 \mathbf{A}+2 \mathbf{B}\)

Short answer

Answer: Yes

Step-by-step solution

Understand the given matrices and equation

We have two matrices A and B: \(\mathbf{A}=\left(\begin{array}{rrr} {3} & {2} & {-1} \\ {2} & {-1} & {2} \\ {1} & {2} & {1} \end{array}\right)\) and \(\mathbf{B}=\left(\begin{array}{rrr} {2} & {1} & {-1} \\ {-2} & {3} & {3} \\ {1} & {0} & {2} \end{array}\right)\) We have to verify the following equation treating both sides as separate calculations: \(2(\mathbf{A}+\mathbf{B})=2 \mathbf{A}+2 \mathbf{B}\)

Calculate \(\mathbf{A}+\mathbf{B}\)

To add two matrices, we add the corresponding elements in each matrix: $\mathbf{A}+\mathbf{B}=\left(\begin{array}{rrr} {3+2} & {2+1} & {-1+(-1)} \\ {2+(-2)} & {-1+3} & {2+3} \\ {1+1} & {2+0} & {1+2} \end{array}\right) =\left(\begin{array}{rrr} {5} & {3} & {-2} \\ {0} & {2} & {5} \\ {2} & {2} & {3} \end{array}\right)$

Calculate \(2(\mathbf{A}+\mathbf{B})\)

To perform scalar multiplication, we multiply each element of the given matrix by the scalar value: $2(\mathbf{A}+\mathbf{B})=2\left(\begin{array}{rrr} {5} & {3} & {-2} \\ {0} & {2} & {5} \\ {2} & {2} & {3} \end{array}\right) =\left(\begin{array}{rrr} {10} & {6} & {-4} \\ {0} & {4} & {...

Key concepts

Matrix Addition

Matrix addition is a fundamental concept in matrix algebra. It involves adding two matrices by summing the corresponding elements from each matrix. This is only possible when the matrices being added are of the same size, meaning they have the same number of rows and columns.

• Each element in the resulting matrix is the sum of the elements from the two matrices in the same position.
• For instance, if you have two matrices, \( \mathbf{A} \) and \( \mathbf{B} \) with dimensions \( m \times n \), their sum \( \mathbf{A} + \mathbf{B} \), will also be \( m \times n \).

This concept is crucial as it forms the basis for more complex matrix operations.

Scalar Multiplication

Scalar multiplication is another vital operation in linear algebra, especially in the context of matrix operations. It involves multiplying each element of a matrix by the same scalar value.

This operation is straightforward but powerful as it allows you to scale a matrix, effectively changing its values uniformly. For example, if you have a matrix \( \mathbf{C} \) and a scalar \( k \), the product \( k \mathbf{C} \) results in a new matrix where each element of \( \mathbf{C} \) is multiplied by \( k \).

• If \( k = 2 \) and \( \mathbf{C} \) is a matrix, each element of \( \mathbf{C} \) will be doubled in \( 2\mathbf{C} \).
• Scalar multiplication does not change the size or shape of a matrix, only its values.

This process is a building block for solving equations involving matrices and for performing other matrix operations efficiently.

Matrix Operations

Matrix operations encompass a variety of procedures that can be carried out on matrices. These operations include addition, subtraction, multiplication, and more advanced functions like finding inverses or determinants.

In the context of the exercise, we deal with addition and scalar multiplication which are relatively straightforward. However, understanding these simple operations is key to grasping more complex matrix algebra tasks.

• Matrix multiplication involves combining two matrices to form a new matrix, observing specific rules related to the arrangement and multiplication of row and column elements.
• The original problem showcases the use of both addition and scalar multiplication to break down and verify expressions.

Matrix operations create a framework that is essential for applications in engineering, physics, computer science, and more.

Linear Algebra

Linear algebra is a branch of mathematics focusing on vector spaces and the linear mappings between them. It uses matrices extensively to represent and solve linear systems of equations.

• Linear algebra is foundational for many areas including computer graphics, machine learning, and optimization, where matrices play a critical role.
• Matrices in linear algebra can be used to transform data, model relationships, and perform calculations that would be difficult or impossible by direct manual computation.

The exercise you are tackling is part of linear algebra's focus on matrix operations, emphasizing the rules that govern how matrices interact. By mastering basic operations like addition and scalar multiplication, students can easily transition to more complex concepts like eigenvectors and transformations, which are vital for higher-level applications.