Perform each matrix row operation and write the new matrix.
$\begin{bmatrix}2 & -6 & 4 & \vert & 10 \\1 & 5 & -5 & \vert & 0 \\3 & 0 & 4 & \vert & 7\end{bmatrix}\quad \frac{1}{2}R_1$

Verified step by step guidance

1. Identify the given matrix and the row operation to be performed. The matrix is: \[\left[ \begin{array}{ccc|c} 2 & -6 & 4 & 10 \\ 1 & 5 & -5 & 0 \\ 3 & 0 & 4 & 7 \end{array} \right]\] and the operation is \[\frac{1}{2} R_1\], which means multiply every element in row 1 by \[\frac{1}{2}\].
2. Apply the operation to row 1 by multiplying each element in the first row by \[\frac{1}{2}\]: - First element: \[2 \times \frac{1}{2} = 1\] - Second element: \[-6 \times \frac{1}{2} = -3\] - Third element: \[4 \times \frac{1}{2} = 2\] - Augmented element: \[10 \times \frac{1}{2} = 5\]
3. Write the new matrix with the updated first row and the unchanged rows 2 and 3: \[\left[ \begin{array}{ccc|c} 1 & -3 & 2 & 5 \\ 1 & 5 & -5 & 0 \\ 3 & 0 & 4 & 7 \end{array} \right]\]
4. Double-check that only row 1 has changed and rows 2 and 3 remain the same.
5. This completes the row operation \[\frac{1}{2} R_1\] on the matrix.

Key Concepts

Matrix Row Operations

Matrix row operations are techniques used to manipulate the rows of a matrix to simplify or solve systems of linear equations. These include row swapping, scaling a row by a nonzero constant, and adding a multiple of one row to another. They preserve the solution set of the system.

Scalar Multiplication of a Row

Scalar multiplication involves multiplying every element in a row by the same nonzero constant. This operation changes the row but keeps the system equivalent. For example, multiplying row 1 by 1/2 scales all its entries by 0.5, simplifying the row for further operations.

Augmented Matrix Representation

An augmented matrix combines the coefficients of variables and constants from a system of linear equations into one matrix. It helps visualize and perform row operations efficiently to solve the system. The vertical bar separates coefficients from constants.