Problem 46 - Precalculus: Mathematical for Calculus

Question

Write the system of equations as a matrix equation (see Example 6). $$\left\\{\begin{aligned} x-y+z &=2 \\ 4 x-2 y-z &=2 \\ x+y+5 z &=2 \\ -x-y-z &=2 \end{aligned}\right.$$

Step-by-Step Solution

Answer

The matrix equation is \[ \begin{bmatrix} 1 & -1 & 1 \\ 4 & -2 & -1 \\ 1 & 1 & 5 \\ -1 & -1 & -1 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 2 \\ 2 \\ 2 \\ 2 \end{bmatrix} \].

Step 1: Identify the Coefficient Matrix

First, we identify the coefficients from each equation to form a coefficient matrix. The system of equations is:1. \( x - y + z = 2 \) 2. \( 4x - 2y - z = 2 \) 3. \( x + y + 5z = 2 \) 4. \( -x - y - z = 2 \) The coefficients of \( x, y, \) and \( z \) can be arranged into the matrix:\[ A = \begin{bmatrix} 1 & -1 & 1 \ 4 & -2 & -1 \ 1 & 1 & 5 \ -1 & -1 & -1 \end{bmatrix} \]

Step 2: Form the Variable Column Matrix

Next, identify the variables of the system and arrange them into a column matrix:\[ X = \begin{bmatrix} x \ y \ z \end{bmatrix} \]

Step 3: Identify the Constant Matrix

Now, identify the constants on the right-hand side of each equation and arrange them into another column matrix:\[ B = \begin{bmatrix} 2 \ 2 \ 2 \ 2 \end{bmatrix} \]

Step 4: Write the System as a Matrix Equation

Finally, combine the matrices in the order: coefficient matrix \( A \), variable matrix \( X \), and constant matrix \( B \) to express the system as a matrix equation:\[ A \cdot X = B \]Or explicitly:\[ \begin{bmatrix} 1 & -1 & 1 \ 4 & -2 & -1 \ 1 & 1 & 5 \ -1 & -1 & -1 \end{bmatrix} \begin{bmatrix} x \ y \ z \end{bmatrix} = \begin{bmatrix} 2 \ 2 \ 2 \ 2 \end{bmatrix} \]

Key Concepts

Understanding the Coefficient Matrix

When dealing with a system of equations, we often aim to represent the system in a succinct form known as a matrix equation. A key component of this is the coefficient matrix. This matrix encapsulates all the numerical coefficients from each equation in the system. To construct the coefficient matrix, we carefully line up the coefficients of each variable from equation to equation.
For instance, let's consider the system:

• Equation 1: \( x - y + z = 2 \)
• Equation 2: \( 4x - 2y - z = 2 \)
• Equation 3: \( x + y + 5z = 2 \)
• Equation 4: \( -x - y - z = 2 \)

From these equations, we gather the coefficients into the matrix:\[A = \begin{bmatrix} 1 & -1 & 1 \4 & -2 & -1 \1 & 1 & 5 \-1 & -1 & -1 \end{bmatrix}\] This matrix effectively organizes and represents the linear relationships in our system.

What is a Column Matrix?

Column matrices, often referred to as column vectors, are matrices with only one column and multiple rows. They are particularly useful when representing variables or constants in a system of equations. In our example, the variables \( x, y, \) and \( z \) can be grouped into a single column matrix or vector. This grouping simplifies the structure of our equations.
For our system, the variable matrix looks like this:\[X = \begin{bmatrix} x \ y \ z \end{bmatrix}\]In parallel, the constants on the right-hand side of the equations also form a column matrix:\[B = \begin{bmatrix} 2 \ 2 \ 2 \ 2 \end{bmatrix}\]These column matrices let us express systems of equations more neatly, ultimately being vital to transforming the intricate written system into solvable matrix forms.

Expressing Systems of Equations as Matrix Equations

Matrix equations provide a streamlined way of expressing and solving systems of equations. By pulling together the coefficient matrix, variable matrix, and constant matrix, we construct a unified equation: \( A \cdot X = B \).This transformation provides a clear pathway to finding solutions.
In our earlier example, everything comes together in the matrix equation:\[\begin{bmatrix} 1 & -1 & 1 \4 & -2 & -1 \1 & 1 & 5 \-1 & -1 & -1 \end{bmatrix} \begin{bmatrix} x \ y \ z \end{bmatrix} = \begin{bmatrix} 2 \ 2 \ 2 \ 2 \end{bmatrix}\]This equation concisely places all the elements of the system together:

• The coefficient matrix \( A \) holds the interplay of the variables with different coefficients.
• The column matrix \( X \) houses the core variables of interest.
• And the constant matrix \( B \) contains all the outputs of each equation.

Solving for \( X \) offers the solution to the entire system, demonstrating the practicality of matrix equations in representing relationships in mathematics.