Problem 49 - Precalculus

Question

Apply Cramer's rule to solve each system of equations, if possible. $$\begin{aligned} x+y-z &=5 \\ x-y+z &=-1 \\ -2 x-2 y+2 z &=-10 \end{aligned}$$

Step-by-Step Solution

Answer

Cramer's rule cannot be applied because the determinant is 0.

Step 1: Identify Coefficients Matrix

Identify the coefficient matrix \( A \) from the system of equations:\[ A = \begin{bmatrix} 1 & 1 & -1 \ 1 & -1 & 1 \ -2 & -2 & 2 \end{bmatrix}. \]

Step 2: Compute Determinant of Coefficients Matrix

Calculate the determinant of the matrix \( A \), denoted as \( \det(A) \): \[ \det(A) = \begin{vmatrix} 1 & 1 & -1 \ 1 & -1 & 1 \ -2 & -2 & 2 \end{vmatrix}. \]Using cofactor expansion along the first row, we have:\[ \det(A) = 1 \cdot \begin{vmatrix} -1 & 1 \ -2 & 2 \end{vmatrix} - 1 \cdot \begin{vmatrix} 1 & 1 \ -2 & 2 \end{vmatrix} - 1 \cdot \begin{vmatrix} 1 & -1 \ -2 & -2 \end{vmatrix}. \]Calculating each 2x2 determinant:- \( \begin{vmatrix} -1 & 1 \ -2 & 2 \end{vmatrix} = (-1)(2) - (1)(-2) = -2 + 2 = 0 \)- \( \begin{vmatrix} 1 & 1 \ -2 & 2 \end{vmatrix} = (1)(2) - (1)(-2) = 2 + 2 = 4 \)- \( \begin{vmatrix} 1 & -1 \ -2 & -2 \end{vmatrix} = (1)(-2) - (-1)(-2) = -2 - 2 = -4 \)Thus, \( \det(A) = 1(0) - 1(4) - 1(-4) = -4 + 4 = 0 \).

Step 3: Determine Solvability

Since the determinant of the coefficient matrix \( A \) is 0, Cramer's rule cannot be applied to solve the system of equations as a unique solution does not exist. The system may either be dependent or inconsistent.

Key Concepts

Determinant

In matrix algebra, the **determinant** is a special number that can be calculated from a square matrix. It gives key insights into the matrix's properties, such as invertibility and solutions in systems of equations. For a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the determinant is calculated as \( ad - bc \). In the case of larger matrices, such as a 3x3, the determinant involves more complex calculations using minors and cofactors.
Understanding the determinant is crucial, especially when you are dealing with systems of linear equations. A determinant of zero indicates that the matrix is **singular**, meaning it has no unique inverse, and the system of equations could be dependent or inconsistent. In simpler terms, if the determinant is zero, you cannot solve the system using Cramer's Rule because a unique solution does not exist.
For the given problem, calculating the determinant of the coefficient matrix revealed that the value is zero. As a result, Cramer's Rule is not applicable, highlighting the significance of checking the determinant before proceeding with matrix solutions.

System of Equations

A **system of equations** is a collection of two or more equations with the same set of unknowns. Solving these systems involves finding the values for each variable that satisfy all equations simultaneously. Systems can be classified into three types: consistent and independent, consistent and dependent, or inconsistent.

- **Consistent and Independent**: A unique solution exists where the equations intersect at a single point. - **Consistent and Dependent**: There are infinitely many solutions, typically when two equations represent the same line or plane. - **Inconsistent**: No solution exists as the equations represent parallel lines or planes that never intersect.
Cramer's Rule, a method for solving a system of linear equations, is applicable when the system is consistent and the coefficient matrix is non-singular (meaning its determinant is not zero). This ensures that each variable has a unique solution. In the provided exercise, the determinant of the matrix was zero, suggesting the system might be dependent or inconsistent. Further analysis would be necessary to determine which is the case.

Matrix Algebra

Matrix algebra is a branch of mathematics that deals with matrices and their operations. Key operations include addition, subtraction, multiplication, and finding a determinant, among others. It simplifies the process of solving systems of linear equations.
In Cramer's Rule, matrix algebra plays a central role. The coefficients of variables in a system of equations form a **coefficient matrix**. This matrix is used to check if the system can be solved for a unique solution. The determinant of this matrix dictates whether Cramer's Rule is viable. If it equals zero, the rule cannot proceed further as highlighted in the original exercise.
These concepts of matrix algebra not only apply to solving equations but also to various fields such as physics, computer science, and engineering, where matrix calculations are prevalent.
By understanding matrix algebra, one can efficiently work with complex systems and make informed decisions about the best methods to find solutions.