In Exercises 13–18, perform each matrix row operation and write the new matrix.

Verified step by step guidance

1. Identify the matrix and the row operation to perform: The matrix is a 3x4 matrix, and the operation is -3R_1 + R_2.
2. Multiply each element of the first row (R_1) by -3: [-3*1, -3*(-3), -3*2, -3*0].
3. Add the result from step 2 to the corresponding elements of the second row (R_2): [3 + (-3*1), 1 + (-3*(-3)), -1 + (-3*2), 7 + (-3*0)].
4. Write the new second row after performing the operation.
5. The new matrix will have the updated second row, while the first and third rows remain unchanged.

Key Concepts

Matrix Row Operations

Matrix row operations are techniques used to manipulate the rows of a matrix to achieve a desired form, often for solving systems of equations. The three primary operations include swapping two rows, multiplying a row by a non-zero scalar, and adding or subtracting a multiple of one row to another. These operations are fundamental in methods like Gaussian elimination.

Row Echelon Form

Row echelon form is a specific arrangement of a matrix where all non-zero rows are above any rows of all zeros, and the leading coefficient of each non-zero row (the first non-zero number from the left) is to the right of the leading coefficient of the previous row. This form is crucial for simplifying matrices and solving linear systems efficiently.

Linear Combinations

A linear combination involves creating a new vector (or row in a matrix) by multiplying existing vectors (or rows) by scalars and adding the results. In the context of the given operation, -3R1 + R2 means taking three times the first row, multiplying it by -1, and adding it to the second row, which is a key step in transforming the matrix.