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

Verified step by step guidance

1. Identify the given matrix and label the rows as Row 1, Row 2, and Row 3.
2. Multiply each element of Row 1 by 2. This is the scalar multiplication step.
3. Add the resulting values from the previous step to the corresponding elements of Row 2. This is the row addition step.
4. Replace the original Row 2 with the new values obtained from the addition.
5. Write down the new matrix with the updated Row 2, while Row 1 and Row 3 remain unchanged.

Key Concepts

Matrix Operations

Matrix operations involve various manipulations of matrices, including addition, subtraction, and scalar multiplication. In this context, row transformations are a specific type of operation where we modify one row of a matrix based on the values of another row. Understanding these operations is crucial for performing tasks such as solving systems of equations or finding the inverse of a matrix.

Row Transformation

Row transformation refers to the process of applying specific operations to the rows of a matrix to achieve a desired form, often used in Gaussian elimination. Common transformations include swapping rows, multiplying a row by a scalar, and adding a multiple of one row to another. These transformations help simplify matrices, making it easier to solve linear equations or analyze the matrix's properties.

Scalar Multiplication

Scalar multiplication is the process of multiplying each entry of a matrix by a constant value, known as a scalar. In the context of the given question, multiplying row 1 by 2 means that every element in that row will be doubled. This operation is fundamental in row transformations, as it allows for the adjustment of rows to facilitate further operations or to achieve specific matrix forms.