Problem 39 - Calculus for Biology and Medicine

Question

Write each system in matrix form. (There is no need to solve the systems). $$ \begin{array}{r} 2 x_{1}+3 x_{2}-x_{3}=0 \\ 3 x_{2}+x_{3}=1 \\ x_{1}-x_{3}=2 \end{array} $$

Step-by-Step Solution

Answer

The system's matrix form is: \[ Ax = b \, \] where \[ A = \begin{bmatrix} 2 & 3 & -1 \\ 0 & 3 & 1 \\ 1 & 0 & -1 \end{bmatrix}, \, x = \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}, \, b = \begin{bmatrix} 0 \\ 1 \\ 2 \end{bmatrix} \].

Step 1: Identify the Coefficients

Examine each equation of the system to identify the coefficients of each variable for every equation. Here, we have the following coefficients:- In the first equation, the coefficients are 2, 3, and -1 for \(x_1, x_2,\) and \(x_3\) respectively.- In the second equation, the coefficients are 0, 3, and 1 for \(x_1, x_2,\) and \(x_3\) respectively.- In the third equation, the coefficients are 1, 0, and -1 for \(x_1, x_2,\) and \(x_3\) respectively.

Step 2: Construct the Coefficient Matrix

Use the coefficients identified in Step 1 to create a matrix, \mathbf{A}\, which represents the coefficients of the system. This results in the following matrix: \[ A = \begin{bmatrix} 2 & 3 & -1 \ 0 & 3 & 1 \ 1 & 0 & -1 \end{bmatrix} \]

Step 3: Write the Variables Vector

Create a column vector for the variables, \mathbf{x}\. This vector consists of the variables \(x_1, x_2, x_3\) of the system: \[ x = \begin{bmatrix} x_1 \ x_2 \ x_3 \end{bmatrix} \]

Step 4: Construct the Result Vector

Construct a column vector, \mathbf{b}\, for the result on the right side of each equation. This results in the vector: \[ b = \begin{bmatrix} 0 \ 1 \ 2 \end{bmatrix} \]

Step 5: Combine into Matrix Form

Combine the coefficient matrix, the variable vector, and the result vector into one matrix equation representing the system:\[ Ax = b \]where \[ \begin{bmatrix} 2 & 3 & -1 \ 0 & 3 & 1 \ 1 & 0 & -1 \end{bmatrix} \begin{bmatrix} x_1 \ x_2 \ x_3 \end{bmatrix} = \begin{bmatrix} 0 \ 1 \ 2 \end{bmatrix} \]

Key Concepts

Coefficient Matrix

In mathematics, a coefficient matrix is an essential part of representing linear systems in a structured way. Each element of the matrix corresponds specifically to the coefficient of a variable from a linear equation. Consider the given system of equations:
\[\begin{align*}2x_1 + 3x_2 - x_3 &= 0, \3x_2 + x_3 &= 1, \x_1 - x_3 &= 2.\end{align*}\]
For each equation, we extract its coefficients to form the coefficient matrix. Each row represents the coefficients from one equation:

• The first equation yields coefficients \(2, 3, -1\).
• The second provides \(0, 3, 1\).
• The third gives \(1, 0, -1\).

Arranging these in rows creates the coefficient matrix \(A\):
\[A = \begin{bmatrix} 2 & 3 & -1 \ 0 & 3 & 1 \ 1 & 0 & -1 \end{bmatrix}\]
This matrix forms the foundation of the matrix representation of the linear system, making it more manageable and systematic to analyze and solve.

Linear Equations

Linear equations form the backbone of many mathematical models and problem-solving techniques. These equations, like those given here, express a relationship in which each term is either a constant or a product of a constant and a variable.
The characteristic feature of linear equations is they graph as straight lines. The general form is:
\[ax_1 + bx_2 + cx_3 = d\]
which represents a plane in three dimensions.
In the example provided:

• The first equation is \(2x_1 + 3x_2 - x_3 = 0\).
• The second is \(3x_2 + x_3 = 1\).
• The third is \(x_1 - x_3 = 2\).

Each equation relates the variables \(x_1, x_2,\) and \(x_3\) using coefficients and results in a specific value. Understanding these helps to formulate problems and their solutions in many areas of mathematics and science.

Vector Representation

Vectors offer an efficient way to express linear systems. Using vectors simplifies the computation and manipulation of linear equations. For the given system, we use three vectors:

• \(\mathbf{x}\), the variables vector, which holds all variables involved in the linear equations.
• \(\mathbf{b}\), the constants or results vector, which comprises the constants on the right-hand side of each equation.
• \(\mathbf{A}\), the previously discussed coefficient matrix, which holds all coefficients of the variables.

In our example, we write:

• \(\mathbf{x} = \begin{bmatrix} x_1 \ x_2 \ x_3 \end{bmatrix}\)
• \(\mathbf{b} = \begin{bmatrix} 0 \ 1 \ 2 \end{bmatrix}\)
• \(\mathbf{A} = \begin{bmatrix} 2 & 3 & -1 \ 0 & 3 & 1 \ 1 & 0 & -1 \end{bmatrix}\)

These vectors are combined into a matrix equation \(A\mathbf{x} = \mathbf{b}\). The beauty of the vector representation lies in its ability to declutter complex systems and make operations such as addition, subtraction, and multiplication more straightforward.