Problem 37 - Precalculus

Question

Perform the indicated row operations on each augmented matrix. $$\left[\begin{array}{rrrr|r} 1 & 0 & 5 & -10 & -5 \\ 0 & 1 & 2 & -3 & -2 \\ 0 & 2 & -3 & 0 & -1 \\ 0 & -3 & 2 & -1 & -3 \end{array}\right] \quad \begin{array}{l} R_{3}-2 R_{2} \rightarrow R_{3} \\ R_{4}+3 R_{2} \rightarrow R_{4} \end{array}$$

Step-by-Step Solution

Answer

The updated matrix is \[ \begin{array}{rrrr|r} 1 & 0 & 5 & -10 & -5 \\ 0 & 1 & 2 & -3 & -2 \\ 0 & 0 & -7 & 6 & 3 \\ 0 & 0 & 8 & -10 & -9 \end{array} \].

Step 1: Identifying the Row Operations

The problem indicates two specific row operations: replacing row 3 ( R_3 ) with the result of R_3 - 2R_2 , and replacing row 4 ( R_4 ) with the result of R_4 + 3R_2 . We will apply these transformations to the given augmented matrix.

Step 2: Applying the Operation on \(R_3\)

Perform the operation R_3 - 2R_2 on row 3: \[\begin{aligned}R_3 &= [0, 2, -3, 0, -1] - 2 imes [0, 1, 2, -3, -2] \ &= [0, 2-2, -3-4, 0+6, -1+4] \ &= [0, 0, -7, 6, 3].\end{aligned}\]Replace row 3 with the new values.

Step 3: Applying the Operation on \(R_4\)

Perform the operation R_4 + 3R_2 on row 4: \[\begin{aligned}R_4 &= [0, -3, 2, -1, -3] + 3 imes [0, 1, 2, -3, -2] \ &= [0, -3+3, 2+6, -1-9, -3-6] \ &= [0, 0, 8, -10, -9].\end{aligned}\]Replace row 4 with the new values.

Step 4: Updated Augmented Matrix

After performing the row operations, the updated augmented matrix is:\[\left[\begin{array}{rrrr|r} 1 & 0 & 5 & -10 & -5 \0 & 1 & 2 & -3 & -2 \0 & 0 & -7 & 6 & 3 \0 & 0 & 8 & -10 & -9 \end{array}\right].\]This is the final matrix after the specified row operations.

Key Concepts

Augmented Matrix

An augmented matrix is a way to represent a system of linear equations using a matrix format. This matrix combines the coefficients of the variables and the constants of the equations into a single structure. Let's break down why it's helpful:

• The left side of the augmented matrix contains the coefficients of the variables. For example, for the system of equations representing two variables, you'd see them as the first few columns in matrix form.
• The rightmost column after the vertical line (|) represents the constants or results from the equations.

This matrix form is compact and efficient for calculations. It simplifies the process of applying mathematical transformations like Gaussian Elimination, making it easier to solve systems of equations. The vertical line in the matrix distinguishes between the variable coefficients and constants but is part of the matrix structure and doesn't impact the calculations.

Gaussian Elimination

Gaussian Elimination is a method for solving systems of linear equations. It involves transforming the augmented matrix into a form where you can easily solve for each variable. Here's how it works:

• First, you aim to form a triangular structure in the matrix. This means making sure that all entries below the main diagonal are zero.
• Next, you use what are known as pivot operations to modify the matrix gradually.

Once the matrix is in its upper triangular form, solving the system becomes straightforward through back substitution. This process is powerful for systems with multiple equations and provides a systematic approach to finding solutions.

Elementary Row Operations

Elementary row operations are the fundamental operations you can perform on the rows of a matrix during methods like Gaussian Elimination. There are three main types of operations:

• Swapping two rows of the matrix.
• Multiplying all elements of a row by a non-zero scalar.
• Adding or subtracting a multiple of one row to another row.

In the provided exercise, these operations were applied to modify the rows of the augmented matrix. For example, subtracting twice the second row from the third one or adding three times the second row to the fourth. Such operations are crucial as they allow you to manipulate the matrix into a more solvable form without changing the solution of the system they represent. These operations maintain equivalence, which means they do not change the solution set of the system of equations.