Problem 42 - Calculus for Biology and Medicine

Question

Write each system in matrix form. $$ \begin{array}{r} x_{1}-2 x_{2}+x_{3}=1 \\ -2 x_{1}+x_{2}-3 x_{3}=0 \end{array} $$

Step-by-Step Solution

Answer

The system in matrix form is \( \begin{bmatrix} 1 & -2 & 1 \\ -2 & 1 & -3 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} = \begin{bmatrix} 1 \\ 0 \end{bmatrix} \).

Step 1: Identify the Variables and Coefficients

The given system of equations is: \( x_1 - 2x_2 + x_3 = 1 \) and \( -2x_1 + x_2 - 3x_3 = 0 \). Identify the coefficients of each variable in each equation. The coefficients of \( x_1, x_2, \) and \( x_3 \) in the first equation are 1, -2, and 1, respectively. In the second equation, they are -2, 1, and -3.

Step 2: Write the Matrix of Coefficients

Arrange the coefficients of the variables in a matrix form. For the given system, the matrix of coefficients is: \[\begin{bmatrix}1 & -2 & 1 \-2 & 1 & -3\end{bmatrix}\] This matrix represents the left side of the equations without the variables.

Step 3: Form the Variable Vector

List the variables in a column vector. For this system, the variable vector is \( \begin{bmatrix} x_1 \ x_2 \ x_3 \end{bmatrix} \).

Step 4: Form the Constant Vector

List the constants from the right side of the equations in a column vector. For the system, the constant vector is \( \begin{bmatrix} 1 \ 0 \end{bmatrix} \).

Step 5: Assemble the System in Matrix Form

Combine the matrix of coefficients, the variable vector, and the constant vector into the standard matrix equation form: \[\begin{bmatrix}1 & -2 & 1 \-2 & 1 & -3\end{bmatrix}\begin{bmatrix}x_1 \x_2 \x_3\end{bmatrix} =\begin{bmatrix}1 \0\end{bmatrix}\] This represents the original system of equations in matrix form.

Key Concepts

Coefficients Matrix

Transforming a system of equations into matrix form begins with creating the coefficients matrix. This matrix includes all the numerical coefficients of the variables from each equation.
For our system of equations:

• In the first equation, the coefficients are 1, -2, and 1 for variables \(x_1\), \(x_2\), and \(x_3\).
• In the second equation, the coefficients are -2, 1, and -3 respectively for \(x_1\), \(x_2\), and \(x_3\).

The coefficients matrix is a neat way to organize these numbers. It is crucial since it holds the key numerical relationships between the variables. Each row corresponds to an equation in the system. Thus for our example, the coefficients matrix is:\[\begin{bmatrix}1 & -2 & 1 \-2 & 1 & -3\end{bmatrix}\]This matrix concisely represents all the "weights" each variable carries in the system of equations.

Variable Vector

Next, we consider the variables themselves and form what is known as the variable vector. While the coefficients matrix captures the numerical aspects, the variable vector highlights the actual variables involved in the equations.

The variable vector is created by listing the variables \(x_1\), \(x_2\), and \(x_3\) in a vertical column:\[\begin{bmatrix}x_1 \x_2 \x_3\end{bmatrix}\]
Why do we use a vector format? It's mainly for ease of operation. In matrix algebra, multiplying a matrix by a vector is a fundamental operation. By aligning variables in this format, it allows the entire system of equations to be succinctly represented and manipulated. Using this structured approach ensures a neat transition from traditional algebraic notation to matrix-driven methods.

Constant Vector

Alongside the coefficients and variable vector, there is a constant vector. It represents the numbers on the other side of the equations—the terms you equate the linear expressions to.

In our given system, the constants or solutions to each equation are 1 and 0. Hence, the constant vector is represented as:\[\begin{bmatrix}1 \0\end{bmatrix}\]
Having a constant vector is integral for forming the complete matrix equation.
When you write a system in matrix form, combining the

• coefficients matrix
• variable vector
• constant vector

allows you to express the system compactly as a matrix equation:\[A \mathbf{x} = \mathbf{b}\]where \(A\) is the coefficients matrix, \(\mathbf{x}\) is the variable vector, and \(\mathbf{b}\) is the constant vector. This provides a powerful way to solve systems of equations using techniques from linear algebra.