Problem 60 - College Algebra

Question

In applying Cramer's Rule, what should you do if \(D=0 ?\)

Step-by-Step Solution

Answer

If the determinant of the coefficient matrix, D, is 0 while applying Cramer's Rule, it means the system doesn't have a unique solution. Therefore Cramer's Rule can't be applied in such a scenario. The system itself could be either dependent or inconsistent.

Step 1: Understanding Cramer's Rule

Cramer's Rule is a method used to solve systems of linear equations by using the determinants of matrices. It involves two main steps: finding the determinant of the coefficients (D) and substituting the column vectors with the constants while calculating the determinants.

Step 2: Consideration when D=0

If the determinant of the coefficient matrix (D) equals 0, it means that the coefficient matrix is singular, i.e., it does not have an inverse. This is because the determinant is a special number that can tell us a lot about a matrix, and one of the things it says is whether the matrix has an inverse. In such a case, Cramer's Rule cannot be applied.

Step 3: Final Analysis

So, if D=0 during the use of Cramer's Rule, it should be concluded that the system of equations has no unique solution or an infinite number of solutions. Further, when D=0, the linear system could be either dependent or inconsistent and to determine which is the case a different method than Cramer's Rule should be utilized.