Problem 42 - Calculus for Biology and Medicine

Question

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

Step-by-Step Solution

Answer

The system can be written in matrix form as \(AX = B\), where \(A = \begin{bmatrix} 1 & -3 & 1 \\ -2 & 1 & -1 \end{bmatrix}\), \(X = \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}\), and \(B = \begin{bmatrix} 1 \\ 0 \end{bmatrix}\).

Step 1: Identify the Coefficients

Begin by identifying the coefficients of each variable in the system of equations. The system consists of two equations, with variables \(x_1\), \(x_2\), and \(x_3\). For the first equation \(x_1 - 3x_2 + x_3 = 1\), the coefficients are 1, -3, and 1. For the second equation \(-2x_1 + x_2 - x_3 = 0\), the coefficients are -2, 1, and -1.

Step 2: Construct the Coefficient Matrix

Write the coefficients in matrix form as a 2x3 matrix, aligning the coefficients of each variable in columns. This matrix, known as the coefficient matrix \(A\), is: \[ A = \begin{bmatrix} 1 & -3 & 1 \ -2 & 1 & -1 \end{bmatrix}\]

Step 3: Create the Variable Vector

Next, express the variables in vector form. The variable vector \(X\) is composed of the variables from the equations: \[ X = \begin{bmatrix} x_1 \ x_2 \ x_3 \end{bmatrix}\]

Step 4: Formulate the Constant Vector

Finally, organize the constants from each equation into a vector, known as the constant vector \(B\): \[ B = \begin{bmatrix} 1 \ 0 \end{bmatrix}\]

Step 5: Combine into Matrix Equation

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

Key Concepts

Coefficient Matrix

In systems of equations, the **coefficient matrix** is a way to organize the numerical factors (coefficients) of each variable. This matrix is structured by placing all the coefficients in a rectangular arrangement of rows and columns. Each row corresponds to one equation, and each column corresponds to one variable from the system.

Let's break it down:
- **Row 1**: Represents the first equation in the system- **Row 2**: Represents the second equation in the system

The arrangement looks like this:
\[A = \begin{bmatrix}1 & -3 & 1 \-2 & 1 & -1\end{bmatrix}\]
This makes it easier to visualize relationships between variables across different equations. When you consistently use a coefficient matrix, you can easily perform matrix operations like addition, multiplication, and finding solutions using various algebraic methods. This organization also simplifies many computational tasks, especially for larger systems of equations.

Variables Vector

The **variables vector** is another critical component when writing a system of equations in matrix form. This vector is crafted from the set of unknown variables in the system. It acts like a list representing all the variables that you're trying to solve.

For the existing system, the variables vector would be:
\[X = \begin{bmatrix}x_1 \x_2 \x_3\end{bmatrix}\]
This one-column matrix organizes your variables in a straightforward manner, aligning with the corresponding coefficients from the coefficient matrix. The setup comes in handy when defining the equation as \(AX = B\), where \(A\) is the coefficient matrix, \(X\) is the variables vector, and \(B\) is the constant vector. By doing this, you get a clear, ordered view of the entire system, making it easier to manipulate and interpret.

Constant Vector

The **constant vector** in a system of equations consolidates all the standalone terms from each equation. These standalone terms typically sit on the right side of the equals sign. Organizing these constants into a vector helps finalize the transition from a set of separate equations to a single, concise matrix equation.

In our specific example, the constant vector is:
\[B = \begin{bmatrix}1 \0\end{bmatrix}\]
The constant vector aligns with the rows of the coefficient matrix and the variables vector. It represents the outcome for each equation based on the given coefficients and unknown variables. When using matrix form, having this vector makes it both simpler and more systematic to solve the equations, as it condenses the process into one elegant mathematical expression. By forming the matrix equation \(AX = B\), calculating solutions becomes a structured process, balancing all elements in a cohesive manner.