Problem 33 - Precalculus

Question

Perform the indicated row operations on each augmented matrix. $$\left[\begin{array}{rrr|r} 1 & -2 & -1 & 3 \\ 2 & 1 & -3 & 6 \\ 3 & -2 & 5 & -8 \end{array}\right] \quad R_{2}-2 R_{1} \rightarrow R_{2}$$

Step-by-Step Solution

Answer

The new second row is \([0, 5, -1, 0]\).

Step 1: Identify the Row Operation

The given row operation is \( R_{2} - 2R_{1} \rightarrow R_{2} \). This operation tells us to replace the second row of the matrix with the result of subtracting two times the first row from the second row.

Step 2: Multiply the First Row

Multiply each element of the first row \( [1, -2, -1, 3] \) by 2, resulting in \( [2, -4, -2, 6] \).

Step 3: Subtract from the Second Row

Subtract the resulting row from the second step \([2, -4, -2, 6]\) from the original second row \([2, 1, -3, 6]\). Perform the subtraction element-wise.

Step 4: Calculate New Second Row

Calculate the new second row by performing the subtraction: \( [2 - 2, 1 + 4, -3 + 2, 6 - 6] = [0, 5, -1, 0] \).

Step 5: Write the New Augmented Matrix

Replace the original second row with the new second row calculated in the previous step. The new augmented matrix is:\[ \left[ \begin{array}{rrr|r} \ 1 & -2 & -1 & 3 \ 0 & 5 & -1 & 0 \ 3 & -2 & 5 & -8 \ \end{array} \right] \]

Key Concepts

Augmented Matrix

An augmented matrix is a convenient way to represent a system of linear equations. It combines the coefficients of the variables and the constants from each equation into a single matrix. This matrix includes a vertical line that separates the two parts, which helps distinguish between coefficients and constants. For example, the augmented matrix given in the problem is:\[\begin{array}{rrr|r} 1 & -2 & -1 & 3 \ 2 & 1 & -3 & 6 \ 3 & -2 & 5 & -8 \end{array}\]Here, each row corresponds to an equation from the system of equations. The entries before the vertical line are coefficients of the variables, and the entries after the line are the constant terms. This format makes it easier to apply matrix operations, such as row reduction, to solve the system of equations.

Matrix Row Reduction

Matrix row reduction is a method used to simplify matrices through a series of row operations. The aim of row reduction is typically to convert the matrix into a simpler form, such as row-echelon form or reduced row-echelon form. By achieving these forms, we can solve systems of linear equations more easily. When performing row reduction, we typically use the following steps:

• Identify the pivot element, which is usually the first non-zero element in a row.
• Use elementary row operations to create zeros below the pivot.
• Move to the next row and repeat the process, creating a triangular form.

In the exercise given, row reduction is used to perform a specific operation: replacing the second row with the result of a subtraction involving the first row. This is part of the process in achieving a form where the solution to the system becomes easily recognizable.

Elementary Row Operations

Elementary row operations are the building blocks of matrix row reduction. They include three types of operations that can be applied to rows of a matrix:

• Swapping two rows.
• Multiplying a row by a non-zero scalar.
• Adding or subtracting a multiple of one row from another row.

These operations are crucial because they allow us to manipulate matrices without changing the solutions to the system of equations they represent. In the problem example, the operation \( R_2 - 2R_1 \rightarrow R_2 \) is an example of the third type of elementary row operation. Here, the second row is updated by subtracting twice the first row (\( 2R_1 \)) from it. This operation simplifies the matrix, bringing it a step closer to a form that's easier to interpret for solving the system of equations.