Problem 37 - Algebra and Trigonometry Real Mathematics, Real People

Question

Determine whether the statement is true or false. Justify your answer. Cramer's Rule cannot be used to solve a system of linear equations when the determinant of the coefficient matrix is zero.

Step-by-Step Solution

Answer

The statement is true. Cramer's Rule cannot be used to solve a system of linear equations when the determinant of the coefficient matrix is zero because the system may not have a unique solution in this case.

Step 1: Analyzing the Statement

Cramer's Rule can be used to solve a system of linear equations by using determinants. According to Cramer's Rule, the system of linear equations has a unique solution if and only if the determinant of the coefficient matrix of the system of equations is not zero.

Step 2: Verifying the Statement

The statement 'Cramer's Rule cannot be used to solve a system of linear equations when the determinant of the coefficient matrix is zero' is evaluated. It aligns with the condition of Cramer's Rule. When the determinant of the coefficient matrix is zero, the system of equations may either have infinite solutions or no solutions, hence Cramer's Rule cannot be used.

Step 3: Statement Evaluation

Given this information, it can be concluded that the statement is true. If the determinant of the coefficient matrix of a system is zero, Cramer's Rule cannot be used to find a unique solution for that system.