MTH120 College Algebra Chapter 7.6

7.6 Solving Systems with Gaussian Elimination

Gaussian Elimination is a systematic method for solving systems of linear equations. It involves performing a sequence of operations on the augmented matrix of the system to transform it into a simpler, triangular form, making it easier to solve for the variables. Here are the steps for solving a system of linear equations using Gaussian Elimination:

Step 1: Augmented Matrix Write the system of equations in the form of an augmented matrix, where the coefficients of the variables and the constants are combined.

Step 2: Row Echelon Form (REF) Perform row operations to transform the augmented matrix into row-echelon form (REF). The main goals in this step are to create leading 1s (1s in the leading coefficients) and zeros below and above those leading 1s.

Start with the first row and first column (left-most), which contains the first variable. Make the first element (the leading coefficient) in the first row equal to 1 by dividing the entire row by the coefficient. This process is called "row scaling."
Use the first row to create zeros below the leading 1 by subtracting multiples of the first row from the rows below it.
Move to the second column and repeat the process, making the leading coefficient in the second row equal to 1 and creating zeros below it.
Continue this process, moving from left to right and top to bottom, until you have a triangular form. The result is the row-echelon form (REF).

Step 3: Reduced Row Echelon Form (RREF) Continue from the row-echelon form to convert it into reduced row-echelon form (RREF):

In the REF, make sure each leading 1 is the only nonzero entry in its column. You can use row operations to achieve this.
Use back-substitution to turn the numbers above the leading 1s into zeros.
The resulting matrix is in RREF, where you have leading 1s and zeros both above and below them.

Step 4: Solve for Variables Read the system of equations from the RREF. For each row, the variable corresponding to the leading 1 can be solved directly, while the other variables become parameters.

Step 5: Interpret Results Interpret the solutions in the context of your problem, if applicable.

Gaussian Elimination is a powerful method for solving systems of linear equations, but it requires careful attention to detail and the application of row operations to manipulate the augmented matrix. It's important to practice this method to become proficient at solving systems of equations efficiently.

To write the augmented matrix of a system of equations, you need to organize the coefficients of the variables and the constants into a matrix format. Here's how you can do it, along with examples:

Step 1: Write the System of Equations

Start with the system of equations you want to represent as an augmented matrix. For example, consider the following system of equations:

\begin{aligned} 2 � + 3 � - � & = 10 \\ 4 � - � + 2 � & = 8 \\ � + 2 � + 3 � & = 15 \end{aligned}

Step 2: Organize the Coefficients

Identify the coefficients of the variables (x, y, z) and the constants on the right-hand side. Arrange them in rows and columns to create a matrix. The coefficients of the variables become the entries in the left portion of the matrix, while the constants form the rightmost column. Here's how the matrix is formed for the example system:

[\begin{array}{ccccc} 2 & 3 & - 1 & ∣ & 10 \\ 4 & - 1 & 2 & ∣ & 8 \\ 1 & 2 & 3 & ∣ & 15 \end{array}]

In this matrix, the vertical bar (|) separates the coefficient matrix on the left from the constant matrix on the right.

The matrix above is the augmented matrix corresponding to the given system of equations.

Example 2:

Let's consider another example:

System of Equations:

\begin{aligned} � - 2 � + 3 � & = 6 \\ 2 � + � + � & = 5 \\ 4 � - 3 � + 2 � & = 10 \end{aligned}

Augmented Matrix:

[\begin{array}{ccccc} 1 & - 2 & 3 & ∣ & 6 \\ 2 & 1 & 1 & ∣ & 5 \\ 4 & - 3 & 2 & ∣ & 10 \end{array}]

This augmented matrix represents the second system of equations.

Creating the augmented matrix is a crucial step in solving systems of equations using various methods, including Gaussian elimination, matrix operations, and linear algebra techniques. Once you have the augmented matrix, you can perform operations on it to solve the system of equations efficiently.

To write an augmented matrix for a system of linear equations, follow these steps:

Step 1: Write the System of Equations

Start with the system of linear equations you want to represent as an augmented matrix. For example, consider the following system:

\begin{aligned} 3 � + 2 � - � & = 5 \\ 2 � - 4 � + 3 � & = 8 \\ � + � + � & = 4 \end{aligned}

Step 2: Identify Coefficients and Constants

Identify the coefficients of the variables (x, y, z) and the constants on the right-hand side of each equation. Arrange these coefficients and constants into a matrix format. The coefficients of the variables become the entries in the left portion of the matrix, while the constants form the rightmost column. Here's how to organize the matrix for the example system:

[\begin{array}{ccccc} 3 & 2 & - 1 & ∣ & 5 \\ 2 & - 4 & 3 & ∣ & 8 \\ 1 & 1 & 1 & ∣ & 4 \end{array}]

In this matrix, the vertical bar (|) separates the coefficient matrix on the left from the constant matrix on the right. This is the augmented matrix corresponding to the given system of equations.

Note: The order of the variables in the matrix should be consistent with the order of the variables in the equations. In this example, the order is x, y, z.

Creating the augmented matrix is an essential step in solving systems of equations using various methods, such as Gaussian elimination, matrix operations, and linear algebra techniques. Once you have the augmented matrix, you can perform operations on it to solve the system of equations efficiently.

To write a system of equations from an augmented matrix, you need to reverse the process of creating the augmented matrix. Here's how to do it, along with examples:

Step 1: Identify the Coefficients and Constants

Given an augmented matrix, identify the coefficients of the variables (x, y, z, etc.) in the left portion of the matrix and the constants on the right side of the vertical bar (|).

Step 2: Write the Equations

Use the coefficients and constants to write the equations, keeping track of the variables. The coefficients become the coefficients in front of the variables, and the constants are the right-hand sides of the equations.

Step 3: Form the System of Equations

Combine the equations to form the system of equations.

Example 1:

Given an augmented matrix:

[\begin{array}{ccccc} 2 & - 1 & 3 & ∣ & 8 \\ 1 & 4 & - 2 & ∣ & 3 \\ - 3 & 2 & 1 & ∣ & - 7 \end{array}]

Step 1: Identify the coefficients and constants:

Coefficients:
- For the first row: 2x, -1y, 3z
- For the second row: 1x, 4y, -2z
- For the third row: -3x, 2y, 1z
Constants:
- For the first row: 8
- For the second row: 3
- For the third row: -7

Step 2: Write the equations:

From the coefficients and constants, we can write the equations:

\begin{aligned} 2 � - � + 3 � & = 8 \\ � + 4 � - 2 � & = 3 \\ - 3 � + 2 � + � & = - 7 \end{aligned}

Step 3: Form the system of equations:

The system of equations is:

\begin{aligned} 2 � - � + 3 � & = 8 \\ � + 4 � - 2 � & = 3 \\ - 3 � + 2 � + � & = - 7 \end{aligned}

This is the system of equations corresponding to the given augmented matrix.

Example 2:

Given an augmented matrix:

[\begin{array}{ccccc} 5 & 1 & 2 & ∣ & 10 \\ 0 & 3 & - 1 & ∣ & 7 \\ 0 & 0 & 4 & ∣ & 8 \end{array}]

Step 1: Identify the coefficients and constants:

Coefficients:
- For the first row: 5x, 1y, 2z
- For the second row: 3y, -1z
- For the third row: 4z
Constants:
- For the first row: 10
- For the second row: 7
- For the third row: 8

Step 2: Write the equations:

From the coefficients and constants, we can write the equations:

\begin{aligned} 5 � + � + 2 � & = 10 \\ 3 � - � & = 7 \\ 4 � & = 8 \end{aligned}

Step 3: Form the system of equations:

The system of equations is:

\begin{aligned} 5 � + � + 2 � & = 10 \\ 3 � - � & = 7 \\ 4 � & = 8 \end{aligned}

This is the system of equations corresponding to the given augmented matrix.

Creating a system of equations from an augmented matrix is essential for solving the system using various methods, including Gaussian elimination, matrix operations, and linear algebra techniques. Once you have the system of equations, you can work on solving it using your preferred method.

Performing row operations on a matrix is a fundamental technique, especially when solving systems of linear equations using methods like Gaussian elimination. There are three primary row operations:

Row Scaling: You can multiply an entire row by a nonzero constant.
Row Addition (or Subtraction): You can add (or subtract) one row to (from) another.
Row Interchange: You can interchange the positions of two rows.

Here are examples of these row operations:

Example 1: Row Scaling

Suppose you have the following 3x4 matrix, and you want to multiply the first row by 2:

[\begin{array}{cccc} 1 & 3 & - 2 & 5 \\ 2 & 4 & 1 & 6 \\ - 1 & 2 & 3 & 7 \end{array}]

Row Scaling (R1 * 2):

[\begin{array}{cccc} 2 & 6 & - 4 & 10 \\ 2 & 4 & 1 & 6 \\ - 1 & 2 & 3 & 7 \end{array}]

You've multiplied the first row by 2.

Example 2: Row Addition (or Subtraction)

Now, let's say you want to add the second row to the third row:

Matrix before the operation:

[\begin{array}{cccc} 2 & 6 & - 4 & 10 \\ 2 & 4 & 1 & 6 \\ - 1 & 2 & 3 & 7 \end{array}]

Row Addition (R3 = R3 + R2):

[\begin{array}{cccc} 2 & 6 & - 4 & 10 \\ 2 & 4 & 1 & 6 \\ 1 & 6 & 4 & 13 \end{array}]

You've added the second row to the third row.

Example 3: Row Interchange

Suppose you want to interchange the positions of the first and third rows:

Matrix before the operation:

[\begin{array}{cccc} 2 & 6 & - 4 & 10 \\ 2 & 4 & 1 & 6 \\ 1 & 6 & 4 & 13 \end{array}]

Row Interchange (R1 ↔ R3):

[\begin{array}{cccc} 1 & 6 & 4 & 13 \\ 2 & 4 & 1 & 6 \\ 2 & 6 & - 4 & 10 \end{array}]

You've swapped the first and third rows.

Row operations are a key part of the Gaussian elimination process for solving systems of linear equations. By applying these operations strategically, you can transform a matrix into row-echelon form and ultimately solve for the variables in the system.

Gaussian Elimination is a method for solving systems of linear equations by transforming the augmented matrix of the system into a simpler, triangular form. The ultimate goal is to perform a sequence of row operations to create leading 1s (1s in the leading coefficients) and zeros below and above those leading 1s. Here's how Gaussian Elimination works with examples:

Step 1: Augmented Matrix

Start with the system of equations, and write the augmented matrix where the coefficients of the variables and constants are combined.

Example 1: Consider the system:

\begin{aligned} 2 � + 3 � - � & = 10 \\ 4 � - � + 2 � & = 8 \\ � + 2 � + 3 � & = 15 \end{aligned}

The augmented matrix is:

[\begin{array}{ccccc} 2 & 3 & - 1 & ∣ & 10 \\ 4 & - 1 & 2 & ∣ & 8 \\ 1 & 2 & 3 & ∣ & 15 \end{array}]

Step 2: Row Echelon Form (REF)

Perform row operations to transform the augmented matrix into row-echelon form (REF). The main goals are to create leading 1s (1s in the leading coefficients) and zeros below and above those leading 1s.

Example 2:

Start with the augmented matrix:

[\begin{array}{ccccc} 2 & 3 & - 1 & ∣ & 10 \\ 4 & - 1 & 2 & ∣ & 8 \\ 1 & 2 & 3 & ∣ & 15 \end{array}]

We'll perform row operations to create leading 1s and zeros:

Row 2 = Row 2 - 2 * Row 1 (to create a zero below the leading 1 in Row 1):

[\begin{array}{ccccc} 2 & 3 & - 1 & ∣ & 10 \\ 0 & - 7 & 4 & ∣ & - 12 \\ 1 & 2 & 3 & ∣ & 15 \end{array}]

Row 3 = Row 3 - (1/2) * Row 1 (to create a zero below the leading 1 in Row 1):

[\begin{array}{ccccc} 2 & 3 & - 1 & ∣ & 10 \\ 0 & - 7 & 4 & ∣ & - 12 \\ 0 & 1.5 & 4.5 & ∣ & 5 \end{array}]

Row 3 = Row 3 - (1.5/7) * Row 2 (to create a zero below the leading 1 in Row 2):

[\begin{array}{ccccc} 2 & 3 & - 1 & ∣ & 10 \\ 0 & - 7 & 4 & ∣ & - 12 \\ 0 & 0 & 3 & ∣ & 9 \end{array}]

Step 3: Reduced Row Echelon Form (RREF)

Continue to convert the REF into reduced row-echelon form (RREF):

Row 2 = Row 2 / -7 (to make the leading coefficient in Row 2 equal to 1):

[\begin{array}{ccccc} 2 & 3 & - 1 & ∣ & 10 \\ 0 & 1 & - 4 / 7 & ∣ & 12 / 7 \\ 0 & 0 & 1 & ∣ & 3 \end{array}]

Row 1 = Row 1 - 3 * Row 3 (to create a zero above the leading 1 in Row 3):

[\begin{array}{ccccc} 2 & 3 & 0 & ∣ & 1 \\ 0 & 1 & - 4 / 7 & ∣ & 12 / 7 \\ 0 & 0 & 1 & ∣ & 3 \end{array}]

Row 1 = Row 1 - 3 * Row 3 (to create zeros above and below the leading 1 in Row 2):

[\begin{array}{ccccc} 2 & 0 & 0 & ∣ & 1 \\ 0 & 1 & 0 & ∣ & 3 \\ 0 & 0 & 1 & ∣ & 3 \end{array}]

Step 4: Solve for Variables

Read the system of equations from the RREF. In this case, it becomes:

\begin{aligned} 2 � & = 1 \\ � & = 3 \\ � & = 3 \end{aligned}

Step 5: Interpret Results

Interpret the solutions in the context of your problem, if applicable. In this example, $� = 1$ , $� = 3$ , and $� = 3$ are the solutions to the system of equations.

Gaussian Elimination is a powerful method for solving systems of linear equations, and it can be applied to larger systems with more variables as well.

Solving a system of linear equations using matrices involves transforming the system into a matrix equation of the form $� � = �$ , where $�$ is the coefficient matrix, $�$ is the vector of variables, and $�$ is the vector of constants. Then, you can solve for $�$ by performing matrix operations. Let's go through an example:

Example:

Consider the following system of linear equations:

\begin{aligned} 2 � + 3 � - � & = 10 \\ 4 � - � + 2 � & = 8 \\ � + 2 � + 3 � & = 15 \end{aligned}

Step 1: Write the Coefficient Matrix and the Constant Vector

Identify the coefficients of the variables (x, y, z) and the constants to form the coefficient matrix $�$ and the constant vector $�$ :

Coefficient Matrix $�$ :

� = [\begin{array}{ccc} 2 & 3 & - 1 \\ 4 & - 1 & 2 \\ 1 & 2 & 3 \end{array}]

Constant Vector $�$ :

� = [\begin{array}{c} 10 \\ 8 \\ 15 \end{array}]

Step 2: Set Up the Matrix Equation

Now, set up the matrix equation $� � = �$ :

[\begin{array}{ccc} 2 & 3 & - 1 \\ 4 & - 1 & 2 \\ 1 & 2 & 3 \end{array}] [\begin{array}{c} � \\ � \\ � \end{array}] = [\begin{array}{c} 10 \\ 8 \\ 15 \end{array}]

Step 3: Solve for $�$

To solve for $�$ , you can use matrix algebra to isolate $�$ :

$� � = �$

Multiply both sides by the inverse of matrix $�$ to isolate $�$ :

$� = �^{- 1} �$

You can use matrix software or a calculator to find the inverse of $�$ and then multiply it by $�$ . The result will be the solution vector $�$ .

Step 4: Calculate the Solution

Let's find the solution by calculating $�^{- 1}$ (the inverse of $�$ ) and then multiplying it by $�$ .

Matrix $�$ is:

� = [\begin{array}{ccc} 2 & 3 & - 1 \\ 4 & - 1 & 2 \\ 1 & 2 & 3 \end{array}]

We'll calculate $�^{- 1}$ using matrix software or a calculator:

[\begin{array}{ccc} 0.25 & 0.75 & - 0.25 \\ 0.5 & - 0.5 & 0.5 \\ - 0.25 & 0.25 & 0.25 \end{array}]

Now, we multiply $�^{- 1}$ by $�$ :

[\begin{array}{ccc} 0.25 & 0.75 & - 0.25 \\ 0.5 & - 0.5 & 0.5 \\ - 0.25 & 0.25 & 0.25 \end{array}] [\begin{array}{c} 10 \\ 8 \\ 15 \end{array}] \approx [\begin{array}{c} 1 \\ 2 \\ 3 \end{array}]

So, the solution to the system of linear equations is $� = 1$ , $� = 2$ , and $� = 3$ .

This is how you can solve a system of linear equations using matrices. The process involves forming the coefficient matrix $�$ , the constant vector $�$ , and solving for $�$ by multiplying the inverse of $�$ with $�$ .

Matrices are widely used in finance for various applications, including portfolio optimization, risk assessment, and financial modeling. Let's look at a specific example of how 3x3 matrices can be applied to finance:

Portfolio Return and Risk Analysis:

Suppose you have a portfolio of three assets: stocks A, B, and C. You want to analyze the expected return and risk of this portfolio. You have historical data for the returns of these assets over a certain period. You can use matrices to perform this analysis.

Let's assume the following information for the expected returns and standard deviations (risk) of the three assets:

Asset A: Expected return = 8%, Standard deviation = 12%
Asset B: Expected return = 10%, Standard deviation = 15%
Asset C: Expected return = 6%, Standard deviation = 10%

Step 1: Formulate the Matrix of Expected Returns:

You can create a 3x1 matrix (column vector) to represent the expected returns of the three assets:

Expected Return Matrix (�) = [\begin{array}{c} 0.08 \\ 0.10 \\ 0.06 \end{array}]

Step 2: Formulate the Covariance Matrix:

You can create a 3x3 matrix (covariance matrix) to represent the pairwise covariances between the three assets. The diagonal elements represent the variances (squared standard deviations), and off-diagonal elements represent covariances.

For simplicity, let's assume the covariances are as follows:

Covariance between A and B: 0.01
Covariance between A and C: 0.005
Covariance between B and C: 0.02

The variances are the squares of the standard deviations.

Covariance Matrix (Σ) = [\begin{array}{ccc} 0.1 2^{2} & 0.01 & 0.005 \\ 0.01 & 0.1 5^{2} & 0.02 \\ 0.005 & 0.02 & 0.1 0^{2} \end{array}]

Step 3: Calculate Portfolio Return and Risk:

Now, if you want to analyze a portfolio with certain weights for each asset, you can represent the weights as a 3x1 matrix (column vector) and calculate the portfolio's expected return and risk using matrix operations.

For example, if the portfolio weights are as follows:

Weight of Asset A: 40%
Weight of Asset B: 30%
Weight of Asset C: 30%

You can represent the weight matrix as:

Weight Matrix (�) = [\begin{array}{c} 0.40 \\ 0.30 \\ 0.30 \end{array}]

The expected return of the portfolio ( $� (�_{�})$ ) can be calculated as:

$� (�_{�}) = �^{�} \cdot �$

Where $�^{�}$ is the transpose of the weight matrix.

The portfolio risk ( $�_{�}$ ) can be calculated as:

$�_{�} = \sqrt{�^{�} \cdot Σ \cdot �}$

Where $Σ$ is the covariance matrix.

Using these formulas and the given matrices, you can calculate the expected return and risk of the portfolio.

This is just one example of how matrices can be applied to finance, specifically in portfolio return and risk analysis. Matrices are used in various other financial calculations and modeling, making them a powerful tool for financial professionals.

Gaussian elimination, a method for solving systems of linear equations, finds practical applications in various real-world scenarios. Here are some examples:

Engineering and Structural Analysis:
- Structural engineers use Gaussian elimination to analyze and design buildings and bridges. They can model the forces acting on structural components and solve for equilibrium conditions.
Electrical Circuit Analysis:
- In electrical engineering, Gaussian elimination is applied to analyze complex electrical circuits. By solving systems of linear equations derived from circuit components, engineers can determine voltage, current, and power distribution.
Economics and Financial Modeling:
- Economists and financial analysts use Gaussian elimination to model economic systems, such as input-output analysis. It can help determine the economic impact of different industries and sectors within an economy.
Optimization Problems:
- Optimization problems, such as linear programming, involve finding the best solution that maximizes or minimizes an objective function while satisfying a set of linear constraints. Gaussian elimination can be used to solve these problems in areas like supply chain management, logistics, and resource allocation.
Data Analysis and Regression Analysis:
- In data analysis, multiple linear regression models involve solving systems of equations to find coefficients that best fit the data. Gaussian elimination can be used to estimate these coefficients and make predictions.
Network Flow Analysis:
- Gaussian elimination can be applied to solve network flow problems, such as determining the most efficient way to transport goods through a network of interconnected nodes and edges. This has applications in transportation and logistics planning.
Scientific Research and Simulation:
- Researchers use Gaussian elimination to solve systems of equations that arise in scientific simulations, modeling physical systems, and predicting outcomes in various fields, from physics to biology.
Image and Signal Processing:
- In image and signal processing, Gaussian elimination can be used for tasks like image reconstruction, filtering, and enhancing features in images and signals. It helps identify patterns and structures in data.
Markov Chain Analysis:
- Gaussian elimination plays a role in analyzing Markov chains, which model systems that transition between states with probabilistic rules. This is used in applications like predicting the behavior of systems with randomness, such as financial markets.
Machine Learning and Deep Learning:
- Some machine learning and deep learning algorithms involve solving systems of equations. Gaussian elimination or related techniques can be used for tasks like solving linear regression problems or optimizing neural network weights.

In each of these real-world applications, Gaussian elimination serves as a powerful tool for solving complex systems of linear equations, making it a valuable technique in various scientific, engineering, and business domains.

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