Problem 72 - Precalculus with Limits

Question

TRUE OR FALSE? In Exercises 71-74, determine whether the statement is true or false. Justify your answer. You cannot use Cramer's Rule when solving a system of linear equations if the determinant of the coefficient matrix is zero.

Step-by-Step Solution

Answer

The statement 'You cannot use Cramer's Rule when solving a system of linear equations if the determinant of the coefficient matrix is zero' is true.

Step 1: Understanding the Problem

The question challenges whether or not it is possible to use Cramer's Rule to solve a system of linear equations when the determinant of its coefficient matrix is zero.

Step 2: Understanding Cramer's Rule

Cramer's Rule is an explicit formula for the solution of a system of linear equations with as many equations as unknowns, possible only when the determinant of the coefficient matrix is not zero.

Step 3: Determining the Truthfulness

Based on the explanation in Step 2, it is clear that Cramer's Rule cannot be applied when the determinant of the coefficient matrix is zero. This is because the rule involves dividing by the determinant of the coefficient matrix, and division by zero is undefined in mathematics.

Key Concepts

Determinant of the Coefficient Matrix

In a system of linear equations, the coefficient matrix is a matrix composed of the coefficients of the variables in the equations. The determinant of this matrix is a special number that can give us important information about the system. Think of the determinant as a kind of 'gatekeeper' that tells us if a unique solution exists.

If the determinant of the coefficient matrix is not zero, it indicates that the linear equations are independent and, therefore, likely have a unique solution. On the other hand, if the determinant is zero, this often means that the system does not have a unique solution. This could lead to infinite solutions or possibly no solution at all. Hence, calculating the determinant is a crucial first step in understanding the behavior of the system.

To compute the determinant, you need to apply specific mathematical operations depending on the size of the matrix. For a 2x2 matrix, it is computed as \( a_{11}a_{22} - a_{12}a_{21} \). For larger matrices, the process involves more complex calculations such as cofactor expansion.

System of Linear Equations

A system of linear equations consists of multiple linear equations that share the same set of variables. These systems can describe real-world situations in fields such as physics, engineering, and economics. Each equation in the system gives information about the relationships between the variables.

The goal when solving a system is to find values for the variables that satisfy all the equations simultaneously. This can result in:

• A unique solution where only one set of values works for all equations.
• Infinite solutions where many sets of values work.
• No solution where no set of values works.

To determine which of these applies, we use methods like substitution, elimination, or matrix techniques like Cramer's Rule, which depends on the determinant of the coefficient matrix being non-zero.

Division by Zero

Division by zero is a concept in mathematics that results in an undefined operation. When you divide by zero, it essentially means you're trying to divide something into zero parts, which makes no sense and cannot be done. This is why on calculators or in mathematical expressions, attempts to divide by zero lead to errors or undefined results.

In the context of solving systems of linear equations using Cramer's Rule, division by zero becomes a concern when the determinant of the coefficient matrix is zero. This is because Cramer's Rule relies on dividing by the determinant to find the solution for each variable. When the determinant equals zero, this operation becomes impossible, rendering Cramer's Rule inapplicable.

Thus, it's crucial to calculate the determinant first when considering Cramer's Rule, to ensure no division by zero will occur. If the determinant is zero, alternative methods must be used to solve the system.