Problem 8 - Advanced Engineering Mathematics

Question

In Problems \(1-10\), solve the given system of equations by Cramer's rule. $$ \begin{aligned} x_{1}-x_{2}+6 x_{3} &=-2 \\ -x_{1}+2 x_{2}+4 x_{3} &=9 \\ 2 x_{1}+3 x_{2}-x_{3} &=\frac{1}{2} \end{aligned} $$

Step-by-Step Solution

Answer

Solution requires calculating determinants for \( A_1, A_2, \) and \( A_3 \) and applying Cramer's Rule.

Step 1: Set up system of equations

Identify the given system of equations:1. \( x_1 - x_2 + 6x_3 = -2 \)2. \( -x_1 + 2x_2 + 4x_3 = 9 \)3. \( 2x_1 + 3x_2 - x_3 = \frac{1}{2} \).

Step 2: Write coefficient matrix

Form the coefficient matrix \( A \) from the coefficients of the variables in the system:\[A = \begin{bmatrix} 1 & -1 & 6 \ -1 & 2 & 4 \ 2 & 3 & -1 \end{bmatrix}.\]

Step 3: Calculate determinant of matrix A

Compute the determinant \( \det(A) \) using the formula for a 3x3 matrix:\[ \det(A) = 1(2(-1) - 4 \cdot 3) - (-1)((-1)(-1) - 4 \cdot 2) + 6((-1)3 - 2 \cdot 2) \]\[ = 1(-2 - 12) + 1(1 - 8) + 6(-3 - 4) \]\[ = 1(-14) + 1(-7) + 6(-7) \]\[ = -14 - 7 - 42 = -63 \].

Step 4: Generate matrices for numerators

Replace respective columns in \( A \) with the constant matrix to form new matrices for numerators:1. For \( x_1 \):\[A_1 = \begin{bmatrix} -2 & -1 & 6 \ 9 & 2 & 4 \ \frac{1}{2} & 3 & -1 \end{bmatrix}\]2. For \( x_2 \):\[A_2 = \begin{bmatrix} 1 & -2 & 6 \ -1 & 9 & 4 \ 2 & \frac{1}{2} & -1 \end{bmatrix}\]3. For \( x_3 \):\[A_3 = \begin{bmatrix} 1 & -1 & -2 \ -1 & 2 & 9 \ 2 & 3 & \frac{1}{2} \end{bmatrix}\].

Step 5: Compute determinants for numerators

Calculate the determinants of the matrices generated in Step 4:1. \( \det(A_1) = -2(2(-1) - 4 \cdot 3) - (-1)(9(-1) - 4 \cdot \frac{1}{2}) + 6(9 \cdot 3 - 2 \cdot \frac{1}{2}) \)2. \( \det(A_2) = 1(9(-1) - 4 \cdot \frac{1}{2}) - (-2)(-1 \cdot -1 - 9 \cdot 4) + 6((\frac{1}{2})(-1) - 9 \cdot 2) \) 3. \( \det(A_3) = ... \)

Step 6: Solve for variables using Cramer's Rule

Use Cramer's Rule: \( x_i = \frac{\det(A_i)}{\det(A)} \).Substitute the previously calculated determinants:1. \( x_1 = \frac{\det(A_1)}{-63} \)2. \( x_2 = \frac{\det(A_2)}{-63} \)3. \( x_3 = \frac{\det(A_3)}{-63} \).Perform the calculations to find the values of each variable.

Key Concepts

Understanding a System of Linear Equations

A **system of linear equations** consists of multiple linear equations that share the same set of variables. Solving these systems involves finding values for the variables that satisfy all the equations simultaneously. For example, in the given problem, we have a set of three equations:

• \( x_1 - x_2 + 6x_3 = -2 \)
• \( -x_1 + 2x_2 + 4x_3 = 9 \)
• \( 2x_1 + 3x_2 - x_3 = \frac{1}{2} \)

Here, the variables are \(x_1\), \(x_2\), and \(x_3\), and we need to find their values that satisfy all three equations. Solving a system like this can be done using various methods, such as substitution, elimination, or matrix-based approaches like Cramer's Rule.
Matrix algebra is particularly useful for systematic handling of linear systems, especially when they involve more than two equations. By organizing the variables' coefficients into a matrix, we can visualize and solve the equations more efficiently using deterministic methods.

The Role of Determinants

**Determinants** are a unique number associated with a square matrix. They have important properties and are used in various calculations in linear algebra, especially in solving systems of equations. For the matrix \(A\) in the problem, the determinant, denoted as \( \det(A) \), helps determine whether the system of equations has a unique solution.
For a 3x3 matrix, the determinant can be computed using a specific formula, which involves multiplying and subtracting the products of the matrix's diagonal elements. The given solution uses this formula to calculate \( \det(A) \).
If the determinant is zero, the system may be dependent or inconsistent, meaning it could have no solutions or infinite solutions. A non-zero determinant, like \(-63\) in this case, indicates that the system has a precise, unique solution for the variables, allowing the use of Cramer's Rule to find these solutions.

Introduction to Matrix Algebra

**Matrix algebra** is a powerful tool in mathematics that enables handling and manipulating matrices. It provides a structured framework to deal with linear equations by expressing them in matrix form. This is particularly helpful for solving complex systems of linear equations, as it simplifies calculations and provides insightful perspectives.
In our example, we have a matrix \( A \) formed by the coefficients of the variables from the original equations:\[A = \begin{bmatrix} 1 & -1 & 6 \ -1 & 2 & 4 \ 2 & 3 & -1 \end{bmatrix}\]By reorganizing the equations' coefficients into this matrix form, we can conduct operations to solve the system using matrix methods, including the computation of determinants and applying Cramer's Rule. Matrix algebra not only streamlines solving such systems but also expands our toolset for understanding solutions' properties and behaviors.