Use the given row transformation to change each matrix as indicated. See Sample 1. < 3x3 Matrix > ; 4 times row 1 added to row 2

Verified step by step guidance

1. Identify the given matrix and label its rows as \(R_1\), \(R_2\), and \(R_3\) for clarity.
2. Understand the row operation: "4 times row 1 added to row 2" means you will multiply each element of \(R_1\) by 4 and then add the result to the corresponding element in \(R_2\).
3. Write the new row 2 as \(R_2' = R_2 + 4 \times R_1\). This means for each element in row 2, calculate \(R_2'[j] = R_2[j] + 4 \times R_1[j]\) where \(j\) indexes the columns.
4. Keep row 1 (\(R_1\)) and row 3 (\(R_3\)) unchanged since the operation only affects row 2.
5. Construct the new matrix with the updated \(R_2'\) and the original \(R_1\) and \(R_3\) to complete the row transformation.

Key Concepts

Elementary Row Operations

Elementary row operations are basic manipulations performed on the rows of a matrix to simplify it 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 and Notation

A matrix is a rectangular array of numbers arranged in rows and columns. Each element is identified by its row and column position. Familiarity with matrix notation helps in accurately performing and describing operations like adding multiples of one row to another.

Row Addition Operation

The row addition operation involves adding a multiple of one row to another row without changing the solution set of the associated system. For example, adding 4 times row 1 to row 2 means multiplying each element of row 1 by 4 and adding it to the corresponding element in row 2, effectively transforming row 2.