Problem 7 - Advanced Engineering Mathematics

Question

Use either Gaussian elimination or Gauss-Jordan elimination to solve the given system or show that no solution exists. \(x_{1}+x_{2}+x_{3}=0\) \(x_{1}+x_{2}+3 x_{3}=0\)

Step-by-Step Solution

Answer

The solution is \(x_1 = t\), \(x_2 = -t\), \(x_3 = 0\), for any real \(t\).

Step 1: Write the system of equations as an augmented matrix

The given system of equations is: \(x_{1}+x_{2}+x_{3}=0\) and \(x_{1}+x_{2}+3x_{3}=0\). This can be written as the augmented matrix:\[\begin{bmatrix} 1 & 1 & 1 & | & 0 \1 & 1 & 3 & | & 0 \end{bmatrix}\]

Step 2: Perform Row 2 operation to create zero in the first column

Subtract Row 1 from Row 2 to eliminate the first element of the second row:\[R_2 = R_2 - R_1 \]This results in:\[\begin{bmatrix}1 & 1 & 1 & | & 0 \0 & 0 & 2 & | & 0 \end{bmatrix}\]

Step 3: Simplify Row 2

Divide the entire Row 2 by 2 to simplify it:\[R_2 = \frac{1}{2}R_2\]This changes Row 2 to:\[\begin{bmatrix}0 & 0 & 1 & | & 0\end{bmatrix}\]

Step 4: Analyze the resulting matrix and derive the solution

The matrix now appears as:\[\begin{bmatrix}1 & 1 & 1 & | & 0\0 & 0 & 1 & | & 0\end{bmatrix}\]From the second row, \(x_3 = 0\). Substituting \(x_3 = 0\) into the first row gives \(x_1 + x_2 = 0\). Set \(x_1 = t\), then \(x_2 = -t\), where \(t\) is a free parameter.

Key Concepts

Augmented Matrix

An augmented matrix is a convenient way to represent a system of linear equations using matrices. Instead of writing out the equations, we arrange the coefficients and constants into a matrix form. This form allows us to apply matrix operations to strategically manipulate the system and find solutions.

For example, consider the system of equations mentioned:

• Equation 1: \(x_1 + x_2 + x_3 = 0\)
• Equation 2: \(x_1 + x_2 + 3x_3 = 0\)

To create an augmented matrix for this, we write the coefficients and constants like so:\[\begin{bmatrix} 1 & 1 & 1 & | & 0 \ 1 & 1 & 3 & | & 0 \end{bmatrix}\]Here, the vertical bar separates the coefficients of the variables from the constants. This format is advantageous because it sets up the framework for performing Gaussian or Gauss-Jordan elimination, facilitating easier solution finding.

System of Equations

A system of equations is a collection of two or more equations with the same set of variables. The goal is typically to find all sets of variable values that satisfy each of the equations in the system.

Our example consists of two linear equations in three variables:

• \(x_1 + x_2 + x_3 = 0\)
• \(x_1 + x_2 + 3x_3 = 0\)

Despite having three variables, you might find either a unique solution, no solution, or infinitely many solutions depending on the number and type of equations. Sometimes, analyzing the relationships between variables is easier when using an augmented matrix and applying elimination methods.

In some cases, we encounter what is termed a 'free' variable — one that can take any value within a given set — leading to infinitely many solutions as other variables depend on it. This often happens when there are more variables than independent equations, creating a situation where the system is underdetermined.

Gauss-Jordan Elimination

Gauss-Jordan elimination is an extension of Gaussian elimination and is used to solve systems of equations, find the inverse of a matrix, or compute the rank of a matrix. This method involves a series of steps aimed at transforming the system's augmented matrix into what we call reduced row-echelon form (RREF).

Key steps in the process include:

• Eliminate variables step-by-step to achieve zeros below and above each leading coefficient, hence every leading variable gets isolated.
• Scale rows as needed, ensuring leading coefficients are one.

When applied to the example system, the final matrix looked like:\[\begin{bmatrix} 1 & 1 & 1 & | & 0 \ 0 & 0 & 1 & | & 0 \end{bmatrix}\]This form gives clear solutions — the equation in the second row indicates \(x_3 = 0\). Replacement into the top row simplifies this further into the form where the effect of free parameters becomes apparent, giving insight into all possible solutions.

Matrix Operations

Matrix operations are fundamental tools in transforming matrices during the process of solving systems of equations, conducting dimension adjustments, or transforming data. The operations often employed include addition, subtraction, scalar multiplication, and multiplication between matrices.

In the context of Gauss-Jordan elimination:

• **Row addition/subtraction**: Combine or eliminate variables systematically.
• **Scalar multiplication**: Adjust rows towards getting a matrix in the desired form — this was applied in our case by simplifying row 2.

During the transformation from the initial augmented matrix to its reduced form, row operations prove invaluable. For instance, the subtraction of one row from another erased entries within specific columns, guiding each variable out of the equation one at a time.
These operations converge towards a simplified matrix that lays bare the inherent structure of solutions the system may possess.