Problem 118 - Calculus Volume 3

Question

True or False? Justify your answer with a proof or a counterexample. The following system of algebraic equations has a unique solution: $$ \begin{array}{l} 6 z_{1}+3 z_{2}=8 \\ 4 z_{1}+2 z_{2}=4 \end{array} $$

Step-by-Step Solution

Answer

False. The system has no solution as it is inconsistent.

Step 1: Write the system in matrix form

We represent the given system of equations in matrix form: \[ \begin{bmatrix} 6 & 3 \ 4 & 2 \end{bmatrix} \begin{bmatrix} z_1 \ z_2 \end{bmatrix} = \begin{bmatrix} 8 \ 4 \end{bmatrix} \].

Step 2: Calculate the determinant of the coefficient matrix

The determinant of the coefficient matrix \( \begin{bmatrix} 6 & 3 \ 4 & 2 \end{bmatrix} \) is computed as \( (6)(2) - (3)(4) = 12 - 12 = 0 \).

Step 3: Analyze determinant result for solution uniqueness

Since the determinant of the coefficient matrix is zero, the system of equations is either dependent (infinitely many solutions) or inconsistent (no solutions), but it cannot have a unique solution.

Step 4: Verify through substitution or row operations

To further investigate the nature of the solutions, try to simplify the system. Divide the first equation by 3: \( 2z_1 + z_2 = \frac{8}{3} \) and the second equation by 2: \( 2z_1 + z_2 = 2 \). Since these simplify to equations with different constants, the system is inconsistent.

Key Concepts

Systems of Linear Equations

A system of linear equations is a collection of one or more equations involving the same set of variables. The goal is to find the values of the variables that satisfy all equations within the system at the same time. In linear algebra, these equations are usually represented in the form of \( ax + by = c \).
There are three main types of solutions that a system might have:

• A unique solution: This occurs when there is exactly one set of values for the variables that satisfies every equation.
• Infinitely many solutions: This happens when there are multiple sets of values that satisfy the system; the equations are typically dependent on each other.
• No solution: This occurs when there is no possible set of values that would satisfy the equations simultaneously.

Determining the type of solution can be accomplished using various methods, such as substitution, elimination, or matrix operations, which provide a systematic way to handle the systems.

Determinant of Matrices

The determinant is a specific number associated with a square matrix. For a 2x2 matrix like \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the determinant can be calculated using the formula \( ad - bc \).
The determinant is a crucial concept in linear algebra because it provides critical information about the matrix. In the context of a system of linear equations, the determinant can tell us:

• If the determinant is non-zero, the matrix is invertible, and the system has a unique solution.
• If the determinant is zero, the matrix is not invertible, leading either to no solution or infinitely many solutions, indicating that the system is either inconsistent or dependent.

Thus, calculating the determinant is often a quick way to assess the nature of the solutions to a system of equations represented by a matrix.

Matrix Representation

Matrix representation is a way to express systems of linear equations in a compact and manageable format. This is done by translating the system of equations into matrix form. A system like \( \begin{array}{l} 6z_{1} + 3z_{2} = 8 \ 4z_{1} + 2z_{2} = 4 \end{array} \) can be represented as:
\[ \begin{bmatrix} 6 & 3 \ 4 & 2 \end{bmatrix} \begin{bmatrix} z_1 \ z_2 \end{bmatrix} = \begin{bmatrix} 8 \ 4 \end{bmatrix} \]
The first matrix is called the coefficient matrix, containing the coefficients of the variables \( z_1 \) and \( z_2 \). The second is the variable matrix, and the third is the constant matrix.
Matrix representation is beneficial because it allows us to use matrix operations to solve the system. This method is more systematic and can be programmed into computational software for solving larger systems efficiently. The determinant, rank, and other properties of the matrix can give quick insights into the solution set of the system.