Problem 3 - Advanced Engineering Mathematics

Question

Solve the given system of equations by Cramer's rule. $$ \begin{aligned} 0.1 x_{1}-0.4 x_{2} &=0.13 \\ x_{1}-\quad x_{2} &=0.4 \end{aligned} $$

Step-by-Step Solution

Answer

The solution is \( x_1 = 0.1 \) and \( x_2 = -0.3 \).

Step 1: Write the System in Matrix Form

Re-write the system of linear equations as a matrix equation of the form \( A\mathbf{x} = \mathbf{b} \), where \( A \) is the coefficient matrix, \( \mathbf{x} \) is the vector of variables, and \( \mathbf{b} \) is the constant vector. Matrix form:\[A = \begin{bmatrix} 0.1 & -0.4 \ 1 & -1 \end{bmatrix}, \quad \mathbf{x} = \begin{bmatrix} x_1 \ x_2 \end{bmatrix}, \quad \mathbf{b} = \begin{bmatrix} 0.13 \ 0.4 \end{bmatrix}\]

Step 2: Calculate the Determinant of Matrix A

Find the determinant of the coefficient matrix \( A \). The determinant \( \text{det}(A) \) for a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \) is calculated as \( ad - bc \).For our matrix:\[\text{det}(A) = (0.1)(-1) - (1)(-0.4) = -0.1 + 0.4 = 0.3\]

Step 3: Calculate Determinant for x1 (det(A1))

Replace the first column of \( A \) with \( \mathbf{b} \) to form matrix \( A_1 \), then find its determinant.Matrix \( A_1 \):\[A_1 = \begin{bmatrix} 0.13 & -0.4 \ 0.4 & -1 \end{bmatrix}\]Determinant of \( A_1 \):\[\text{det}(A_1) = (0.13)(-1) - (-0.4)(0.4) = -0.13 + 0.16 = 0.03\]

Step 4: Calculate Determinant for x2 (det(A2))

Replace the second column of \( A \) with \( \mathbf{b} \) to form matrix \( A_2 \), then find its determinant.Matrix \( A_2 \):\[A_2 = \begin{bmatrix} 0.1 & 0.13 \ 1 & 0.4 \end{bmatrix}\]Determinant of \( A_2 \):\[\text{det}(A_2) = (0.1)(0.4) - (1)(0.13) = 0.04 - 0.13 = -0.09\]

Step 5: Calculate Solutions for x1 and x2 Using Cramer's Rule

Apply Cramer's Rule to solve for the variables. For \( x_1 \), it's calculated as \( \frac{\text{det}(A_1)}{\text{det}(A)} \), and for \( x_2 \) as \( \frac{\text{det}(A_2)}{\text{det}(A)} \).Thus,\[x_1 = \frac{0.03}{0.3} = 0.1\]\[x_2 = \frac{-0.09}{0.3} = -0.3\]

Key Concepts

Determinant Calculation

A determinant is a special number that can be calculated from a square matrix. For a 2x2 matrix, it is relatively simple to find the determinant. Consider a matrix of the form:

• \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \)

The determinant, often denoted as \( \text{det}(A) \), is calculated using the formula:

• \( \text{det}(A) = ad - bc \)

This value plays a crucial role in solving linear equations using methods like Cramer's Rule. It helps determine whether a system of equations has a unique solution. If the determinant is zero, the system may not have a unique solution. In cases where the determinant is non-zero, as with our example \( \text{det}(A) = 0.3 \), Cramer's Rule can be effectively applied.

Matrix Form

Matrix form is a way of representing a system of linear equations as a matrix equation of the form \( A\mathbf{x} = \mathbf{b} \). Here:

• \( A \) is the coefficient matrix containing the coefficients of the variables in the equations.
• \( \mathbf{x} \) is the vector of variables we want to solve for.
• \( \mathbf{b} \) is the constant vector that contains the values on the right side of the equations.

By rewriting our system of equations in matrix form, we simplify the solution process. The system automatically becomes compact and clear. In our example, the matrix form is:

• \[ A = \begin{bmatrix} 0.1 & -0.4 \ 1 & -1 \end{bmatrix}, \quad \mathbf{x} = \begin{bmatrix} x_1 \ x_2 \end{bmatrix}, \quad \mathbf{b} = \begin{bmatrix} 0.13 \ 0.4 \end{bmatrix} \]

Here, each element of matrix \( A \) matches the coefficients from the original equations. The goal then becomes finding \( \mathbf{x} \) using methods like Cramer's Rule.

Linear System of Equations

A linear system of equations is a collection of one or more linear equations involving the same set of variables. For example:

• \( 0.1x_1 - 0.4x_2 = 0.13 \)
• \( x_1 - x_2 = 0.4 \)

These equations are linear because each term is either a constant or the product of a constant with a single variable. Linear equations graph as straight lines, and a solution to the system represents the point(s) where these lines intersect.
Such systems can be solved using various methods, including substitution, elimination, and Cramer's Rule. Cramer's Rule offers a systematic way to solve for the variables in a linear system when the number of equations matches the number of unknowns and the determinant of the coefficient matrix is non-zero. Understanding how to represent these equations in matrix form, calculate determinants, and apply rules like Cramer's helps in solving and analyzing these systems efficiently.