Problem 19 - Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

The given system cannot be solved using Cramer's Rule as the determinant is zero.

Step 1: Write the system in matrix form

The given system of equations can be rewritten in matrix form as follows:\[A \cdot \begin{pmatrix} x \ y \ z \end{pmatrix} = \begin{pmatrix} 4 \ 6 \ -1 \end{pmatrix}\]Where matrix \(A\) is:\[A = \begin{pmatrix} 1 & -1 & 2 \ 3 & -2 & 4 \ 2 & -2 & 4 \end{pmatrix}\]And the column vector on the right-hand side is the constants from the equations.

Step 2: Calculate the determinant of matrix A

The determinant of matrix \(A\) is denoted as \(\det(A)\). To find it, use the formula for a 3x3 matrix:\[\det(A) = 1((-2)(4) - (4)(-2)) - (-1)((3)(4) - (4)(2)) + 2((3)(-2) - (-2)(2))\]Simplify each part:\[= 1(-8 + 8) + 1(12 - 8) + 2(-6 + 4)\]\[= 1(0) + 1(4) + 2(-2)\]\[= 0 + 4 - 4 = 0\]Since \(\det(A) = 0\), the matrix is singular, and Cramer's rule cannot be applied.

Step 3: Conclusion about uniqueness of solution

Since \(\det(A) = 0\), the coefficient matrix is singular, meaning there may be no unique solution to the system. Cramer's rule is specifically used when there is a unique solution, hence in this case, we cannot determine a solution using it.

Key Concepts

System of Equations

A system of equations is a collection of two or more equations with the same set of variables. In this exercise, the system consists of three equations with three variables: \ \(x - y + 2z = 4\), \ \(3x - 2y + 4z = 6\), \ \(2x - 2y + 4z = -1\). \ Each equation represents a plane in three-dimensional space, and the solution to the system is the point where all planes intersect.

There are different methods to solve a system of equations:

• Graphs: Visually represent the equations to find their intersection.
• Substitution: Solve one equation for a variable, then substitute that into other equations.
• Elimination: Add or subtract equations to eliminate variables sequentially.
• Matrix Methods: Use matrices and algorithms, such as Cramer's Rule, to find solutions.

In our case, we're interested in using matrices, which helps in handling multiple variables simultaneously.

Matrix Determinant

The determinant of a matrix gives important information about the matrix, especially concerning its invertibility. For a square matrix, the determinant is a scalar value that can be calculated using specific formulas depending on the matrix size.

In this exercise, we have a 3x3 coefficient matrix \(A\): \ \[ A = \begin{pmatrix} 1 & -1 & 2 \ 3 & -2 & 4 \ 2 & -2 & 4 \end{pmatrix}\]
The determinant is calculated by:

• Choosing any row or column.
• Multiplying each element by the determinant of the 2x2 matrix that remains after removing the element's row and column.
• Applying specific signs based on checkerboard patterns (i.e., (-1) power of the sum of the row and column indices).

In our matrix \(A\), the calculation reveals that \( ext{det}(A) = 0\). This result tells us something special about the matrix, specifically concerning its singularity.

Singular Matrix

A singular matrix is a type of square matrix that cannot be inverted. This occurs when its determinant is zero. In simpler terms, there is no unique solution to the linear system that can be achieved through traditional methods like Cramer's Rule.

When a matrix is singular, some scenarios are possible:

• The system might have infinite solutions, meaning all equations represent the same plane or lie on a line.
• The system could be inconsistent, implying no solution exists due to some conflict, like parallel planes that never intersect.

For the exercise in question, since \( ext{det}(A) = 0\), matrix \(A\) is singular. Thus, we cannot use Cramer's Rule here as it only applies to systems where there is one unique solution (non-zero determinant). Instead, other techniques, such as Gaussian elimination or parameterization, might be considered to analyze the system further.