Problem 9 - Differential Equations and Linear Algebra

Question

},\( of the given system. $$\begin{aligned}x_{1}+2 x_{2}-… # Determine the coefficient matrix, \)A\(, the right-hand side vector, \)\mathbf{b},\( and the augmented matrix, \)A^{\\#},$ of the given system. $$\begin{aligned}x_{1}+2 x_{2}-3 x_{3} &=1 \\\2 x_{1}+4 x_{2}-5 x_{3} &=2 \\\7 x_{1}+2 x_{2}-x_{3} &=3\end{aligned}$$

Step-by-Step Solution

Answer

The coefficient matrix, \(A\), right-hand side vector, \(\mathbf{b}\), and the augmented matrix, \(A^{\#}\), for the given system are: $$A = \begin{pmatrix}1 & 2 & -3\\2 & 4 & -5\\7 & 2 & -1\end{pmatrix}, \quad \mathbf{b} = \begin{pmatrix}1\\2\\3\end{pmatrix}, \quad A^{\#} = \left[\begin{array}{ccc|c}1 & 2 & -3 & 1\\2 & 4 & -5 & 2\\7 & 2 & -1 & 3\end{array}\right]$$

Step 1: Identify the coefficient matrix

The coefficient matrix, \(A\), consists of the coefficients of the variables in the system of linear equations. In this case, the coefficients are the numbers next to \(x_1\), \(x_2,\) and \(x_3\): $$A = \begin{pmatrix}1 & 2 & -3\\2 & 4 & -5\\7 & 2 & -1\end{pmatrix}$$

Step 2: Identify the right-hand side vector

The right-hand side vector, \(\mathbf{b}\), consists of the constants on the right-hand side of each equation. From the given system of equations, we can see that the constants are: $$\mathbf{b} = \begin{pmatrix}1\\2\\3\end{pmatrix}$$

Step 3: Create the augmented matrix

To create the augmented matrix, \(A^{\#}\), we simply append the right-hand side vector, \(\mathbf{b}\), to the coefficient matrix, \(A\). In this case, we join the two matrices with a vertical line to separate them: $$A^{\#} = \left[\begin{array}{ccc|c}1 & 2 & -3 & 1\\2 & 4 & -5 & 2\\7 & 2 & -1 & 3\end{array}\right]$$ Now, we have the coefficient matrix, \(A\), the right-hand side vector, \(\mathbf{b}\), and the augmented matrix, \(A^{\#}\), for the given system of linear equations.

Key Concepts

Coefficient Matrix

When working with a system of linear equations, the coefficient matrix is a fundamental concept to grasp. It is essentially a compact way to represent the system's structure, focusing solely on the coefficients of the variables without bothering with the constants on the right-hand side of the equations.
For example, take the following system of equations:

• \(x_1 + 2x_2 - 3x_3 = 1\)
• \(2x_1 + 4x_2 - 5x_3 = 2\)
• \(7x_1 + 2x_2 - x_3 = 3\)

The coefficient matrix \(A\) is built using the numbers directly in front of each variable in every equation. So, for this case:\[A = \begin{pmatrix} 1 & 2 & -3 \ 2 & 4 & -5 \ 7 & 2 & -1 \end{pmatrix}\]The matrix filters out the constants and gives a quick overview of how each variable is weighted across the different equations in the system. Understanding the coefficient matrix allows one to apply various algebraic techniques on the system more effectively.

Right-Hand Side Vector

The right-hand side vector, denoted as \(\mathbf{b}\), complements the coefficient matrix by taking into account the constants that appear on the right-hand side of each equation in a system of linear equations. This vector embodies the targets or results that each equation wants to achieve.
In our existing example:

• First equation's constant is \(1\)
• Second equation's constant is \(2\)
• Third equation's constant is \(3\)

We represent these constants in the right-hand side vector like so:\[\mathbf{b} = \begin{pmatrix} 1 \ 2 \ 3 \end{pmatrix}\]Each entry in \(\mathbf{b}\) aligns with the corresponding equation, exhibiting what the left-hand side should equate to after substituting in the solutions. By analyzing \(\mathbf{b}\), one can better grasp the expectations from the system and how the solution set fits those criteria.

System of Linear Equations

A system of linear equations consists of multiple linear equations that share the same set of variables. The overall goal is often to find the values of these variables that satisfy all equations simultaneously.
For example, consider this system:

• \(x_1 + 2x_2 - 3x_3 = 1\)
• \(2x_1 + 4x_2 - 5x_3 = 2\)
• \(7x_1 + 2x_2 - x_3 = 3\)

By examining such a system, we attempt to explore a solution or multiple solutions that hold true for each individual equation.
To solve this, techniques such as substitution, elimination, or using matrices method - including the augmented matrix - often come in handy. The augmented matrix helps by compactly merging the coefficient matrix and the right-hand side vector into one single matrix:\[A^{\#}=\left[\begin{array}{ccc|c}1 & 2 & -3 & 1 \2 & 4 & -5 & 2 \7 & 2 & -1 & 3\end{array}\right]\]This format is particularly powerful in linear algebra as it facilitates various matrix operations like row reduction, simplifying the solution process for the system.