Problem 42 - Advanced Engineering Mathematics

Question

Write the system of equations $$ \begin{aligned} 2 x_{1}+6 x_{2}+x_{3} &=7 \\ x_{1}+2 x_{2}-x_{3} &=-1 \\ 5 x_{1}+7 x_{2}-4 x_{3} &=9 \end{aligned} $$ as a matrix equation \(\mathbf{A} \mathbf{X}=\mathbf{B}\), where \(\mathbf{X}\) and \(\mathbf{B}\) are column vectors.

Step-by-Step Solution

Answer

Matrix equation: \(\mathbf{A} \mathbf{X} = \mathbf{B}\) where \(\mathbf{A} = \begin{bmatrix} 2 & 6 & 1 \\ 1 & 2 & -1 \\ 5 & 7 & -4 \end{bmatrix}\), \(\mathbf{X} = \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}\), \(\mathbf{B} = \begin{bmatrix} 7 \\ -1 \\ 9 \end{bmatrix}\).

Step 1: Identify Components of Matrix Equation

The matrix equation \( \mathbf{A} \mathbf{X} = \mathbf{B} \) involves three components: \( \mathbf{A} \) (the coefficient matrix), \( \mathbf{X} \) (the variable column vector), and \( \mathbf{B} \) (the column vector of constants).

Step 2: Form the Coefficient Matrix \(\mathbf{A}\)

To form the coefficient matrix \( \mathbf{A} \), write down the coefficients of \( x_1, x_2, \) and \( x_3 \) from each equation in separate rows. Thus, \( \mathbf{A} \) becomes \(\begin{bmatrix}2 & 6 & 1 \1 & 2 & -1 \5 & 7 & -4\end{bmatrix}\).

Step 3: Form the Variable Column Vector \(\mathbf{X}\)

The variable vector \( \mathbf{X} \) represents the unknowns and is written as a column vector: \(\begin{bmatrix}x_1 \x_2 \x_3\end{bmatrix}\).

Step 4: Form the Constant Column Vector \(\mathbf{B}\)

The column vector \( \mathbf{B} \) contains the constants from the right-hand side of each equation, forming: \(\begin{bmatrix}7 \-1 \9\end{bmatrix}\).

Step 5: Combine into Matrix Equation

Put together the matrices and vectors to form the complete matrix equation. The system is written as:\[\begin{bmatrix}2 & 6 & 1 \1 & 2 & -1 \5 & 7 & -4\end{bmatrix}\begin{bmatrix}x_1 \x_2 \x_3\end{bmatrix}= \begin{bmatrix}7 \-1 \9\end{bmatrix}\]

Key Concepts

Linear Algebra

Linear algebra is a branch of mathematics that focuses on vectors, vector spaces, linear transformations, and systems of linear equations. It's a powerful tool in various fields such as engineering, computer science, and economics.

Understanding linear algebra is crucial because it provides a systematic method for solving systems of equations. It allows us to work with complex data and perform operations like rotation, stretching, and compressing in a multi-dimensional space.

• It forms the foundation for more advanced topics in mathematics.
• Applications include computer graphics, machine learning, and statistics.

Grasping these concepts can enable one to handle complex mathematical models and computational tasks efficiently.

System of Equations

A system of equations refers to a set of equations with multiple variables. The challenge is to find the values of these variables that satisfy all the equations simultaneously.

When solving these systems, you can utilize methods like substitution, elimination, and using matrices. By transforming the system into a matrix equation, we can employ linear algebra techniques to find solutions more effectively.

• A system can be either linear or non-linear, though the matrix method primarily applies to linear systems.
• Graphical and algebraic methods can sometimes prove solutions.

The matrix equation approach simplifies large systems, making it easier to work with multiple variables at once.

Coefficient Matrix

The coefficient matrix is a matrix that contains only the coefficients of the variables in a system of linear equations. For the given exercise, the coefficients are extracted to form matrix \( \mathbf{A} \).

This matrix essentially represents the relationship between the variables; each row corresponds to an equation, and each column corresponds to a variable. In a mathematical sense, it is the core of the equation that guides the transformation process.

• Matrix operations such as addition, multiplication, and finding the determinant rely heavily on the structure of the coefficient matrix.
• Properties of matrices, such as invertibility, directly affect the solvability of the system.

Understanding the coefficient matrix is critical in forming and solving matrix equations.

Column Vector

A column vector is simply a matrix with a single column. In linear algebra, column vectors are used to represent variables and constants separately in a matrix equation.

In the context of the matrix equation \( \mathbf{A} \mathbf{X} = \mathbf{B} \):

• \( \mathbf{X} \) is the column vector for variables, representing \( x_1, x_2, \) and \( x_3 \).
• \( \mathbf{B} \) is the column vector for constants, representing the values on the right-hand side of each equation.

The use of column vectors is vital as it structurally separates different elements of the equations, making them easier to manipulate and solve using matrix operations.