Problem 18 - Intermediate Algebra

Question

For Problems \(11-30\), use Cramer's rule to find the solution set of each system. (Objective 2) $$ \left(\begin{array}{r} x-2 y+3 z=1 \\ 2 x+y+z=4 \\ 4 x-3 y+7 z=6 \end{array}\right) $$

Step-by-Step Solution

Answer

The system does not have a unique solution because \( \det(A) = 0 \).

Step 1: Write down the coefficient matrix

The system of equations can be expressed as the following coefficient matrix: \[ A = \begin{pmatrix} 1 & -2 & 3 \ 2 & 1 & 1 \ 4 & -3 & 7 \end{pmatrix} \] We will also need the constants matrix: \[ extbf{b} = \begin{pmatrix} 1 \ 4 \ 6 \end{pmatrix} \]

Step 2: Calculate the determinant of the coefficient matrix

Using the matrix \( A \), the determinant \( \det(A) \) is calculated as follows: \[ \det(A) = 1(1 \cdot 7 - 1 \cdot (-3)) - (-2)(2 \cdot 7 - 1 \cdot 4) + 3(2 \cdot (-3) - 1 \cdot 4) \] Evaluating these expressions gives: \[ = 1(7 + 3) - (-2)(14 - 4) + 3(-6 - 4) \] \[ = 10 + 20 - 30 \] \[ = 0 \] Since the determinant is zero, the system doesn't have a unique solution.

Step 3: Explain the result related to Cramer's Rule

Cramer's Rule is only applicable when \( \det(A) eq 0 \). Since \( \det(A) = 0 \), the system either has no solutions or infinitely many solutions. Therefore, we cannot apply Cramer's Rule directly to find the solution set in this case.

Key Concepts

Determinant of a Matrix

The determinant of a matrix is a special number that can help us understand many properties of a matrix, especially when solving systems of equations. For a square matrix, like the one we have in the original problem, the determinant represents a value calculated from its elements. It's often used to determine if a matrix is invertible, which means there's a unique way to solve the matrix equations involving it.
To find the determinant of a 3x3 matrix, we apply a formula that involves the elements of the matrix. For example, the determinant of the coefficient matrix \( A \) is calculated using the rule of Sarrus or cofactor expansion. Once we apply the formula to each element and its corresponding minor, we sum these values, considering the signs.
However, in our case, the determinant is zero: \( \det(A) = 0 \). This result has significant implications. A zero determinant means the matrix is "singular," signifying problems in solving the equations, such as dependency among equations or no intersection points. In simple terms, it tells us that something is off with the system in terms of having a unique solution.

Coefficient Matrix

The coefficient matrix is derived from the coefficients of the variables in a system of linear equations. For any given system, this matrix acts as the foundation for determining solutions using various mathematical tools, including Cramer's Rule.
In the presented system of equations, the coefficients of \( x \), \( y \), and \( z \) form this matrix:

• The first row \((1, -2, 3)\) comes from the equation \( x - 2y + 3z = 1 \).
• The second row \((2, 1, 1)\) relates to the equation \( 2x + y + z = 4 \).
• The third row \((4, -3, 7)\) is formed from \( 4x - 3y + 7z = 6 \).

The coefficient matrix is crucial because it directly influences the existence of solutions and their uniqueness. If the determinant of this matrix is non-zero, we can use methods like Cramer's Rule to find solutions; however, as seen in this exercise, a zero determinant leads to indefinite conclusions, pointing to dependent or coinciding equations.

Systems of Equations

Systems of equations consist of multiple equations working together. These equations typically involve the same set of variables and aim to find a solution where all equations hold true simultaneously. Working with systems can mean either finding an exact solution, noting when no solution exists, or identifying infinitely many solutions.
When solving these systems, we usually express them in matrix form. This choice makes it easier to use various computational techniques. Sometimes, as with our original problem, traditional algebraic methods might not directly work due to issues like a determinant of zero. A matrix-based approach can often reveal the relationships between the equations:

• If the determinant is non-zero, a unique solution typically exists.
• Zero determinants often indicate special cases like parallel planes in three-dimensional systems, meaning there may be either no solutions or infinite solutions depending on if they intersect with a line.

Understanding these systems broadens our ability to approach complex problems, as we develop skills to relate algebraic and geometric insights to find or conclude about solutions.