Problem 39 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Write the given system of linear equations in matrix form. \(\begin{aligned}-x_{1}+x_{2}+x_{3} &=0 \\ 2 x_{1}-x_{2}-x_{3} &=2 \\\\-3 x_{1}+2 x_{2}+4 x_{3} &=4 \end{aligned}\)

Step-by-Step Solution

Answer

The given system of linear equations can be written in matrix form as: \[ \begin{pmatrix} -1 & 1 & 1 \\ 2 & -1 & -1 \\ -3 & 2 & 4 \end{pmatrix} \begin{pmatrix} x_{1} \\ x_{2} \\ x_{3} \end{pmatrix} = \begin{pmatrix} 0 \\ 2 \\ 4 \end{pmatrix} \]

Step 1: Identify the coefficient matrix

Find the matrix formed by the coefficients from the system of linear equations. Here, the coefficient matrix will be a 3x3 matrix. In this case, the coefficient matrix \(\displaystyle A\) is: \[ A= \begin{pmatrix} -1 & 1 & 1 \\ 2 & -1 & -1 \\ -3 & 2 & 4 \end{pmatrix} \]

Step 2: Identify the variable matrix

Now create the matrix containing the variables \(\displaystyle x_{1}\), \(\displaystyle x_{2}\), and \(\displaystyle x_{3}\). This matrix should be a 3x1 matrix. Here, the variable matrix \(\displaystyle \boldsymbol{x}\) is: \[ \boldsymbol{x}= \begin{pmatrix} x_{1} \\ x_{2} \\ x_{3} \end{pmatrix} \]

Step 3: Identify the constant matrix

Lastly, identify the matrix containing the constants from the system of linear equations. This matrix will also be a 3x1 matrix. In this case, the constant matrix \(\displaystyle \boldsymbol{b}\) is: \[ \boldsymbol{b}= \begin{pmatrix} 0 \\ 2 \\ 4 \end{pmatrix} \]

Step 4: Write the system in matrix form

Arrange the matrices in the following way: \(\displaystyle A\boldsymbol{x}=\boldsymbol{b}\). This is the matrix form of the given system of linear equations. Therefore, the system of linear equations in matrix form is: \[ \begin{pmatrix} -1 & 1 & 1 \\ 2 & -1 & -1 \\ -3 & 2 & 4 \end{pmatrix} \begin{pmatrix} x_{1} \\ x_{2} \\ x_{3} \end{pmatrix} = \begin{pmatrix} 0 \\ 2 \\ 4 \end{pmatrix} \]

Key Concepts

Coefficient Matrix

In the realm of linear algebra, understanding the matrix form of a system of equations is very important. It all starts with the coefficient matrix. This matrix forms the backbone of our system of linear equations. The coefficient matrix contains all the coefficients of the variables from each equation. For instance, if you have a system with equations like:\(\begin{aligned}-x_{1}+x_{2}+x_{3} &=0 \2 x_{1}-x_{2}-x_{3} &=2 \-3 x_{1}+2 x_{2}+4 x_{3} &=4 \end{aligned}\)The coefficient matrix, usually denoted as \(A\), looks like this:

• The first row \((-1, 1, 1)\) consists of coefficients from the first equation.
• The second row \((2, -1, -1)\) from the second, and so forth.

Thus, in this example, the coefficient matrix is:\[A= \begin{pmatrix} -1 & 1 & 1 \2 & -1 & -1 \-3 & 2 & 4 \end{pmatrix}\]This structure offers a neat way to keep track of all variables and their respective coefficients. It plays a fundamental role in simplifying and solving systems of equations efficiently.

Variable Matrix

The variable matrix is another essential component in the matrix form of linear equations. This matrix, commonly represented by \(\boldsymbol{x}\), stores all the variables of the equations in question. By putting the variables together in a column, the variable matrix maintains the order and position found in the original equations. Consider this 3x1 matrix:

• The first entry is \(x_{1}\), representing the first variable.
• \(x_{2}\) follows it, and \(x_{3}\) is third.

For example, our system of equations involves variables \(x_{1}, x_{2},\) and \(x_{3}\), which constitute the variable matrix as follows:\[\boldsymbol{x}= \begin{pmatrix} x_{1} \x_{2} \x_{3} \end{pmatrix}\] This matrix plays an important part in bringing structure to the problem. When you multiply the coefficient matrix \(A\) by \(\boldsymbol{x}\), it effectively reconstructs the linear equations, maintaining clarity and order.

Constant Matrix

In any system of linear equations, the constants determine each equation's outcome. These values are grouped into a separate matrix, called the constant matrix, typically denoted \(\boldsymbol{b}\). This matrix collects all the constants from the right-hand side of the equations and aligns them in a single, clear column. For the system examined here, the constants are 0, 2, and 4, producing the constant matrix:\[\boldsymbol{b}= \begin{pmatrix} 0 \2 \4 \end{pmatrix}\]

• The first entry, 0, corresponds to the first equation's right-hand side.
• The second and third, 2 and 4, relate to the second and third equations, respectively.

The constant matrix helps pair and identify the results each equation aims to achieve. It is highly valuable when aligning or solving linear systems because it completes the matrix equation \(A\boldsymbol{x}=\boldsymbol{b}\). The complete set allows us to utilize matrix operations or computational tools to solve for the unknown variables efficiently and accurately.