Problem 40 - Algebra and Trigonometry

Question

Write the system of equations as a matrix equation (see Example 6). $$ \left\\{\begin{aligned} 6 x-y+z &=12 \\ 2 x\quad \quad+z &=7 \\ y-2 z &=4 \end{aligned}\right. $$

Step-by-Step Solution

Answer

The matrix equation is \( \mathbf{A} \mathbf{x} = \mathbf{b} \) with \( \mathbf{A} = \begin{bmatrix} 6 & -1 & 1 \\ 2 & 0 & 1 \\ 0 & 1 & -2 \end{bmatrix} \), \( \mathbf{x} = \begin{bmatrix} x \\ y \\ z \end{bmatrix} \), \( \mathbf{b} = \begin{bmatrix} 12 \\ 7 \\ 4 \end{bmatrix} \).

Step 1: Identifying the Coefficients

To write the system of equations as a matrix equation, first identify the coefficients of each variable in the system: \(6x - y + z = 12\), \(2x + z = 7\), and \(y - 2z = 4\). The coefficients are 6, -1, 1 for the first equation, 2, 0, 1 for the second equation, and 0, 1, -2 for the third equation.

Step 2: Writing the Coefficient Matrix

The next step is to arrange these coefficients into a matrix, known as the coefficient matrix. This matrix will have the rows and columns that correspond to the coefficients of \(x\), \(y\), and \(z\) from each equation. Therefore, the coefficient matrix is:\[\begin{bmatrix}6 & -1 & 1 \2 & 0 & 1 \0 & 1 & -2 \\end{bmatrix}\]

Step 3: Writing the Variable Matrix

Now, write a column matrix for the variables \(x\), \(y\), and \(z\). This matrix is:\[\begin{bmatrix}x \y \z \\end{bmatrix}\]

Step 4: Writing the Constant Matrix

Identify the constants from each equation and write them in a column matrix. The constants are 12, 7, and 4, respectively. Therefore, the constant matrix is:\[\begin{bmatrix}12 \7 \4 \\end{bmatrix}\]

Step 5: Combining into a Matrix Equation

Combine the coefficient matrix, the variable matrix, and the constant matrix into a single matrix equation. The matrix equation looks like this:\[\begin{bmatrix}6 & -1 & 1 \2 & 0 & 1 \0 & 1 & -2 \\end{bmatrix}\begin{bmatrix}x \y \z \\end{bmatrix}=\begin{bmatrix}12 \7 \4 \\end{bmatrix}\]

Key Concepts

Coefficient Matrix

The coefficient matrix is a fundamental part of solving matrix equations. It is essentially a mathematical representation containing only the coefficients of the variables from the system of equations.

Each row in the coefficient matrix corresponds to one equation from the system, and each column corresponds to a specific variable across all equations. This way of organizing coefficients helps us see complex systems in a compact form.

For our exercise, the matrix captures the coefficients for variables \(x\), \(y\), and \(z\) as follows:

• First Equation: Coefficients are 6, -1, and 1, representing 6\(x\), -\(y\), and \(z\).
• Second Equation: Coefficients are 2, 0, and 1, corresponding to 2\(x\), absent \(y\), and \(z\).
• Third Equation: Coefficients are 0, 1, and -2, representing absent \(x\), \(y\), and -2\(z\).

This results in the coefficient matrix:\[\begin{bmatrix}6 & -1 & 1 \2 & 0 & 1 \0 & 1 & -2\end{bmatrix}\]By organizing the coefficients in this manner, we prepare the system for matrix operations such as multiplication.

Variable Matrix

The variable matrix is a straightforward concept that organizes the variables of the system into a neat column matrix. This compact form is crucial when formulating matrix equations.

In our exercise, the purpose of the variable matrix is to list the variables \(x\), \(y\), and \(z\) as a single vertical column. This allows us to easily perform matrix multiplications with the coefficient matrix. Here's what it looks like:

• Variables are stacked vertically,
• This column contains all variables from the system,
• Each variable aligns with its respective coefficient in the coefficient matrix.

The variable matrix for our exercise appears as:\[\begin{bmatrix}x \y \z\end{bmatrix}\]This arrangement is pivotal for linking each variable to its respective equation in the overall matrix equation. It brings structure by visually separating the problem's variables from their coefficients and constants.

Constant Matrix

The constant matrix gathers the constant terms from each equation of the system into a single column matrix. These constants are the numerical values on the opposite side of the equality sign in each equation.

For example, in our system of equations:

• The first equation's constant is 12,
• The second equation's constant is 7,
• The third equation's constant is 4.

We organize these constants into the constant matrix as follows:\[\begin{bmatrix}12 \7 \4\end{bmatrix}\]This constant matrix plays a crucial role in forming the complete matrix equation. The positioning of each constant aligns with the order of the respective equations, thus maintaining the balance of the overall system. By having these constants organized neatly, we can visualize the entire equation in matrix form, resulting in:\[\begin{bmatrix}6 & -1 & 1 \2 & 0 & 1 \0 & 1 & -2\end{bmatrix}\begin{bmatrix}x \y \z\end{bmatrix}=\begin{bmatrix}12 \7 \4\end{bmatrix}\] This matrix structure allows us to utilize various mathematical techniques for solving systems of equations, further emphasizing why these matrices are foundational to understanding matrix equations.