Perform each matrix row operation and write the new matrix.
$\(\begin{bmatrix}\)1 & -3 & 2 & \(\vert\) & 0 \\3 & 1 & -1 & \(\vert\) & 7 \\2 & -2 & 1 & \(\vert\) & 3\(\end{bmatrix}\]\quad\) -3R_1 + R_2$

Verified step by step guidance

1. Identify the given matrix and the row operation to be performed. The matrix is: \[\left[\begin{array}{ccc|c} 1 & -3 & 2 & 0 \\ 3 & 1 & -1 & 7 \\ 2 & -2 & 1 & 3 \end{array}\right]\] and the row operation is \[-3R_1 + R_2\], which means we will multiply the first row by -3 and add it to the second row.
2. Multiply each element of the first row by -3: \[-3 \times R_1 = [-3, 9, -6, 0]\]
3. Add the result from step 2 to the corresponding elements of the second row: \[R_2 + (-3R_1) = [3 + (-3), 1 + 9, -1 + (-6), 7 + 0]\]
4. Calculate each element of the new second row: \[[0, 10, -7, 7]\]
5. Write the new matrix after the row operation, keeping the first and third rows unchanged and replacing the second row with the new row: \[\left[\begin{array}{ccc|c} 1 & -3 & 2 & 0 \\ 0 & 10 & -7 & 7 \\ 2 & -2 & 1 & 3 \end{array}\right]\]

Key Concepts

Matrix Row Operations

Matrix row operations are fundamental techniques used to manipulate matrices to solve systems of linear equations. These include swapping rows, multiplying a row by a scalar, and adding a multiple of one row to another. In this problem, the operation -3R1 + R2 means multiplying the first row by -3 and adding it to the second row.

Augmented Matrices

An augmented matrix represents a system of linear equations, combining the coefficient matrix and the constants into one matrix. This format simplifies applying row operations to solve the system. The vertical bar separates the coefficients from the constants, indicating the system's equations.

Applying Scalar Multiplication and Addition to Rows

To perform the operation -3R1 + R2, multiply each element of the first row by -3, then add the result to the corresponding element in the second row. This process updates the second row while leaving other rows unchanged, helping to simplify the matrix toward row-echelon form.