Problem 42 - Precalculus Mathematics for Calculus

Question

Write the system of equations as a matrix equation. $$\left\\{\begin{aligned} x-y+z &=2 \\ 4 x-2 y-z &=2 \\ x+y+5 z &=2 \\ -x-y-z &=2 \end{aligned}\right.$$

Step-by-Step Solution

Answer

The matrix equation is \( A \cdot \mathbf{x} = \mathbf{b} \), where \( A = \begin{bmatrix} 1 & -1 & 1 \\ 4 & -2 & -1 \\ 1 & 1 & 5 \\ -1 & -1 & -1 \end{bmatrix}, \mathbf{x} = \begin{bmatrix} x \\ y \\ z \end{bmatrix}, \mathbf{b} = \begin{bmatrix} 2 \\ 2 \\ 2 \\ 2 \end{bmatrix} \).

Step 1: Identify Coefficients and Constants

For each equation, identify the coefficients of the variables \(x\), \(y\), and \(z\), and the constant term on the right side of the equation.

Step 2: Form the Coefficient Matrix

Arrange the coefficients into a matrix. For our system, the coefficient matrix is: \[ \begin{bmatrix} 1 & -1 & 1 \ 4 & -2 & -1 \ 1 & 1 & 5 \ -1 & -1 & -1 \end{bmatrix} \]

Step 3: Construct the Variable Column Matrix

Create a column matrix for the variables \(x\), \(y\), and \(z\) as \[ \begin{bmatrix} x \ y \ z \end{bmatrix} \]

Step 4: Form the Constant Matrix

Form a column matrix from the constants on the right side of the equations: \[ \begin{bmatrix} 2 \ 2 \ 2 \ 2 \end{bmatrix} \]

Step 5: Write the Matrix Equation

Combine the coefficient matrix, variable matrix, and constant matrix into a single matrix equation of the form: \[ \begin{bmatrix} 1 & -1 & 1 \ 4 & -2 & -1 \ 1 & 1 & 5 \ -1 & -1 & -1 \end{bmatrix} \cdot \begin{bmatrix} x \ y \ z \end{bmatrix} = \begin{bmatrix} 2 \ 2 \ 2 \ 2 \end{bmatrix} \]

Key Concepts

Coefficient Matrix

In a system of equations, the coefficient matrix is a collection of all the coefficients of the variables. Each row in the matrix corresponds to an equation in the system, and each column aligns with a variable (e.g., \(x\), \(y\), or \(z\)).
The matrix clearly organizes all the coefficients into a structured layout, making it easier to perform various operations like matrix multiplication and row reduction.
For example, in the given system:

• First equation coefficients: (1, -1, 1)
• Second equation coefficients: (4, -2, -1)
• Third equation coefficients: (1, 1, 5)
• Fourth equation coefficients: (-1, -1, -1)

These coefficients are used in forming the coefficient matrix: \[\begin{bmatrix} 1 & -1 & 1 \ 4 & -2 & -1 \ 1 & 1 & 5 \ -1 & -1 & -1 \end{bmatrix} \] This matrix represents the left-hand side of each equation in the system.

Variable Matrix

The variable matrix, also known as the variable column matrix, is a simple and organized way to represent the variables in your system of equations.
For the system involving variables like \(x\), \(y\), and \(z\), these variables are stacked in a single column.
This matrix forms an essential part of the matrix equation.
In our given scenario, it looks like this: \[\begin{bmatrix} x \ y \ z \end{bmatrix} \] Placing variables in a matrix form allows them to be conveniently multiplied or otherwise processed with the coefficient and constant matrices.
This setup is analogous to listing down all the unknowns you want to solve for in a clean, singular form.
It systematically arranges the unknown variable terms.

Constant Matrix

In matrix equations, the constant matrix contains the constants from the right-hand side of each equation in the system.
Just like the variable matrix, the constant matrix is a column matrix, but instead of containing the unknown variables, it holds the known values each equation equals.
This gives it an important role in calculations involving matrix equations.
In our example, the constant values from each equation are all equal to 2. So, our constant matrix is: \[\begin{bmatrix} 2 \ 2 \ 2 \ 2 \end{bmatrix} \] This matrix helps to simplify systems of equations, making analysis and solutions easier to handle.
It acts as a compact way to represent the outcomes that the linear combinations of variables aim to achieve.
By comparing it directly to other matrix operations, it aligns the goal values in a concise format.

System of Equations

A system of equations refers to a set of equations with multiple variables that are all interlinked.
The challenge is to find the value of each variable that satisfies all equations simultaneously.
Matrix equations offer a robust way to solve these systems.
By representing a system of equations as a matrix equation like \[\begin{bmatrix} 1 & -1 & 1 \ 4 & -2 & -1 \ 1 & 1 & 5 \ -1 & -1 & -1 \end{bmatrix} \cdot \begin{bmatrix} x \ y \ z \end{bmatrix} = \begin{bmatrix} 2 \ 2 \ 2 \ 2 \end{bmatrix} \] we connect coefficients and constants directly to the variables.

• The left-hand side (coefficient matrix times variable matrix) represents the linear combinations of the variables.
• The right-hand side (constant matrix) is what these combinations must equal.

Using matrix operations like row reduction, substitution, or using computational tools, these matrices are manipulated to find the values of \(x\), \(y\), and \(z\) that solve the entire system.