Question

Perform the given row operations on \(A,\) where $$A=\left[\begin{array}{ccc}2 & -1 & 7 \\ 0 & 4 & -2 \\ 5 & 0 & 3\end{array}\right]$$ $$\frac{1}{2} R_{2} \rightarrow R_{2}$$

Short answer

The updated matrix is \( \begin{bmatrix} 2 & -1 & 7 \\ 0 & 2 & -1 \\ 5 & 0 & 3 \end{bmatrix} \).

Step-by-step solution

Understand the row operation

The row operation provided is \( \frac{1}{2} R_2 \rightarrow R_2 \). This means we should divide every element in the second row by 2 and replace the original second row with these new values.

Perform the operation

Take each element in the second row of matrix \( A \) which is \( [0, 4, -2] \), and divide each by 2. Calculate: - \( \frac{1}{2} \times 0 = 0 \) - \( \frac{1}{2} \times 4 = 2 \) - \( \frac{1}{2} \times -2 = -1 \)

Write the new matrix

After performing the row operation, the new second row is \( [0, 2, -1] \). Therefore, the modified matrix is: \[ \begin{bmatrix} 2 & -1 & 7 \ 0 & 2 & -1 \ 5 & 0 & 3 \end{bmatrix} \]

Key concepts

Elementary Row Operations

Elementary row operations are essential techniques in linear algebra used to manipulate matrices to make them easier to analyze and solve. They include three main types of operations:
- Row swapping: Exchanging two rows of a matrix.
- Row multiplication: Multiplying all elements of a row by a non-zero constant.
- Row addition: Adding a multiple of one row to another row.
When applying these operations, the elementary row operation does not change the solutions to a system of linear equations represented by the matrix. In the given exercise, the operation \( \frac{1}{2} R_2 \rightarrow R_2 \) is a type of row multiplication. Here, every element in the second row of the matrix \( A \) is divided by 2. Each of these operations is reversible, allowing flexibility in equations manipulation.

Matrix Transformation

Matrix transformation is a change made to a matrix using operations like addition, multiplication, or more complex processes. The goal of transforming a matrix can vary from simplifying it to solving a system of linear equations or finding an inverse.
In our example, we performed a simple transformation by applying \( \frac{1}{2} R_2 \rightarrow R_2 \), which scaled down the elements of the second row by a factor of 2. This kind of transformation can help to:

• Create row-echelon form or reduced row-echelon form which simplifies solving equations.
• Find determinants or inverses more easily.
• Understand linear mappings represented by matrices.

Matrix transformations are crucial for preprocessing matrices in various calculations, as they often lead to simpler forms that are easier to interpret or solve.

Linear Algebra Basics

Linear Algebra is a branch of mathematics dealing with vector spaces and linear mappings between such spaces. Key components learners should be familiar with include vectors, matrices, determinants, and vector spaces.
- **Vectors:** Data points that can be understood as coordinates in a space. They can be represented as either rows or columns. - **Matrices:** Rectangular arrays of numbers, symbols, or expressions arranged in rows and columns. - **Determinants:** Scalar values derived from a square matrix, used to solve linear equations, among other applications. - **Vector spaces:** Collections of vectors that can be scaled and added together.
Understanding these basics is crucial as they form the backbone of systems of linear equations, matrix transformations, and operations. Mastering these topics helps in fields like computer graphics, engineering, physics, and more, where linear systems often arise.