Use the given row transformation to change each matrix as indicated.

Verified step by step guidance

1. Identify the given matrix and label its rows as \( R_1 \), \( R_2 \), and \( R_3 \).
2. Understand the row operation: "2 times row 1 added to row 2" means you will multiply each element of \( R_1 \) by 2 and then add the result to the corresponding element in \( R_2 \).
3. Write the operation mathematically as: \( R_2 \rightarrow R_2 + 2 \times R_1 \).
4. Perform the multiplication of each element in \( R_1 \) by 2, then add these values to the corresponding elements in \( R_2 \) to get the new \( R_2 \).
5. Keep \( R_1 \) and \( R_3 \) unchanged, and write the new matrix with the updated \( R_2 \).

Key Concepts

Elementary Row Operations

Elementary row operations are basic manipulations performed on the rows of a matrix to simplify or solve systems of equations. These include row swapping, scaling a row by a nonzero constant, and adding a multiple of one row to another. Understanding these operations is essential for matrix transformations and solving linear systems.

Matrix Representation of Systems

A matrix can represent a system of linear equations, where each row corresponds to an equation and each column to a variable or constant term. Manipulating the matrix through row operations helps find solutions or simplify the system without changing its solution set.

Row Addition Operation

The row addition operation involves adding a multiple of one row to another row. For example, '2 times row 1 added to row 2' means multiplying row 1 by 2 and adding the result to row 2, replacing row 2. This operation helps eliminate variables and simplify matrices during solving.