Problem 40 - Calculus for Biology and Medicine

Question

Write each system in matrix form. (There is no need to solve the systems). $$ \begin{array}{r} 2 x_{2}-x_{1}=x_{3} \\ 4 x_{1}+x_{3}=7 x_{2} \\ x_{2}-x_{1}=x_{3} \end{array} $$

Step-by-Step Solution

Answer

The system in matrix form is \[ \begin{bmatrix} -1 & 2 & -1 \\ 4 & -7 & 1 \\ -1 & 1 & -1 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} =\begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix} \].

Step 1: Identify Coefficients for Matrix

Examine each equation to determine the coefficients of each variable for the system. For the equation \(2x_2 - x_1 = x_3\), the coefficients are \(-1\) for \(x_1\), \(2\) for \(x_2\), and \(-1\) for \(x_3\). For the equation \(4x_1 + x_3 = 7x_2\), the coefficients are \(4\) for \(x_1\), \(-7\) for \(x_2\), and \(1\) for \(x_3\). The equation \(x_2 - x_1 = x_3\) gives coefficients \(-1\) for \(x_1\), \(1\) for \(x_2\), and \(-1\) for \(x_3\).

Step 2: Write the Coefficient Matrix

Using the coefficients identified in Step 1, write the matrix with each row corresponding to an equation. The coefficient matrix is:\[\begin{bmatrix}-1 & 2 & -1 \4 & -7 & 1 \-1 & 1 & -1\end{bmatrix}\]

Step 3: Write the Variable Matrix

Represent the variables as a column matrix. Since the system involves the variables \(x_1\), \(x_2\), and \(x_3\), the variable matrix is:\[\begin{bmatrix}x_1 \x_2 \x_3\end{bmatrix}\]

Step 4: Write the Constant Matrix

Analyze the right side of each equation to determine the constant terms. In this case, there are no constant terms, as each equation equates to a variable expression; thus, it can be represented as:\[\begin{bmatrix}0 \0 \0\end{bmatrix}\]

Step 5: Assemble the Matrix Equation

Combine the matrices from Steps 2, 3, and 4 to write the matrix form of the system of equations. This matrix equation is written as:\[\begin{bmatrix}-1 & 2 & -1 \4 & -7 & 1 \-1 & 1 & -1\end{bmatrix}\begin{bmatrix}x_1 \x_2 \x_3\end{bmatrix}=\begin{bmatrix}0 \0 \0\end{bmatrix}\]

Key Concepts

Coefficient Matrix

The coefficient matrix is a fundamental part of representing a system of linear equations in matrix form. It consists solely of the coefficients of the variables from each equation. This matrix simplifies the algebraic problem into a more structured format that can be easily manipulated using matrix operations. To construct a coefficient matrix, each row corresponds to an equation from the system, and each column corresponds to a specific variable.
Let's break this down further:

• Identify the coefficients for each variable in every equation. For example, in the equation \(2x_2 - x_1 = x_3\), the coefficients are \(-1\) for \(x_1\), \(2\) for \(x_2\), and \(-1\) for \(x_3\).
• Arrange these coefficients into rows. With three equations from our original exercise, your coefficient matrix looks like this:\[\begin{bmatrix}-1 & 2 & -1 \4 & -7 & 1 \-1 & 1 & -1\end{bmatrix}\]

The primary purpose of the coefficient matrix is organizing the system so you can apply matrix-solving techniques, such as Gaussian elimination or using matrix inverses, once all elements are clearly and correctly positioned.

Variable Matrix

The variable matrix, also known as the column matrix, is a simplified way to represent the unknowns in our system of equations. By arranging variables in a vertical format, we ensure consistency and make the matrix operation more straightforward when multiplied by the coefficient matrix.
To create the variable matrix:

• List all the variables involved in the equations. In our exercise, the involved variables are \(x_1\), \(x_2\), and \(x_3\).
• Order these variables in a single column, matching the order used in the coefficient matrix. Our variable matrix becomes:\[\begin{bmatrix}x_1 \x_2 \x_3\end{bmatrix}\]
• This structure simplifies computations, as the arrangement aligns directly with the columns of the coefficient matrix. Such alignment is crucial in matrix multiplication, where each element is combined across rows and columns to ensure a proper equation setup.

The variable matrix serves as a bridge, linking the format of the coefficient matrix to the solutions sought in typical algebraic processes.

Constant Matrix

The constant matrix is another crucial element in putting a system of equations into matrix form, although it behaves a little differently depending on the nature of the equations. The constant matrix consists of the terms on the right side of the equations, usually representing constant values. In scenarios where each equation in the system matches a sum of terms instead of constants, the constant matrix merely reflects zeroes.
Here's what to do:

• Examine the right-hand side of each equation. In our example, each equation ends with a variable expression, not a numeric constant. Thus, it leads directly to a zero being placed in each entry of the constant matrix.
• Write these across a single-column matrix. For our provided exercises, the constant matrix results in:\[\begin{bmatrix}0 \0 \0\end{bmatrix}\]
• This zero-format simplifies the problem, as the constant-only nature ensures variables equate directly to the values expressed with the coefficient matrix.

In essence, the constant matrix communicates constraints or specific outputs you aim to resolve for in the matrix equation, effectively tying the system together into a solvable entity.