Question

Define elementary row operations and give an example.

Short answer

Elementary row operations are transformations that can be applied to matrices or systems of linear equations. There are three types: row swapping, row scaling, and row addition. For example, given the matrix \(A = \begin{bmatrix} 1 & 4 & 0 \\ 2 & 5 & -1 \\ 3 & 7 & -2 \end{bmatrix}\), we can swap the first and third rows to create \(A' = \begin{bmatrix} 3 & 7 & -2 \\ 2 & 5 & -1 \\ 1 & 4 & 0 \end{bmatrix}\), scale the second row by 2 to create \(A' = \begin{bmatrix} 1 & 4 & 0 \\ 4 & 10 & -2 \\ 3 & 7 & -2 \end{bmatrix}\), or replace the third row with the sum of itself and -3 times the first row to create \(A' = \begin{bmatrix} 1 & 4 & 0 \\ 2 & 5 & -1 \\ -3 & -5 & -2 \end{bmatrix}\).

Step-by-step solution

Define Elementary Row Operations

Elementary row operations are simple operations that can be performed on a matrix or a system of linear equations. There are three types of elementary row operations: 1. Row swapping: In this operation, we can swap or interchange the positions of any two rows in the matrix. 2. Row scaling: In this operation, a non-zero constant is multiplied with all elements of a row. This operation effectively scales the row by the constant. 3. Row addition: In this operation, a row is replaced by the sum of itself and a multiple of another row.

Example of Row Swapping Operation

Given the following matrix: \[ A = \begin{bmatrix} 1 & 4 & 0 \\ 2 & 5 & -1 \\ 3 & 7 & -2 \end{bmatrix} \] We can swap the first and third rows: \[ A' = \begin{bmatrix} 3 & 7 & -2 \\ 2 & 5 & -1 \\ 1 & 4 & 0 \end{bmatrix} \]

Example of Row Scaling Operation

Using the original matrix A: \[ A = \begin{bmatrix} 1 & 4 & 0 \\ 2 & 5 & -1 \\ 3 & 7 & -2 \end{bmatrix} \] We can scale the second row by a constant, say, 2: \[ A' = \begin{bmatrix} 1 & 4 & 0 \\ 4 & 10 & -2 \\ 3 & 7 & -2 \end{bmatrix} \]

Example of Row Addition Operation

Using the original matrix A again: \[ A = \begin{bmatrix} 1 & 4 & 0 \\ 2 & 5 & -1 \\ 3 & 7 & -2 \end{bmatrix} \] We can replace the third row with the sum of itself and the first row multiplied by -3: \[ (-3 * Row_1) + Row_3 = -3 \begin{bmatrix} 1 & 4 & 0 \end{bmatrix} + \begin{bmatrix} 3 & 7 & -2 \end{bmatrix} = \begin{bmatrix} -3 & -5 & -2 \end{bmatrix} \] The matrix after the row addition operation becomes: \[ A' = \begin{bmatrix} 1 & 4 & 0 \\ 2 & 5 & -1 \\ -3 & -5 & -2 \end{bmatrix} \]