Question

When using Cramer's Rule to solve a system of equations, if the determinant of the coefficient matrix equals zero and neither numerator determinant is zero, then the system _______.
A. has infinite solutions
B. is inconsistent
C. is dependent
D. is consistent

Answer 1

The answer would be A. When using Cramer's Rule to solve a system of equations, if the determinant of the coefficient matrix equals zero and neither numerator determinant is zero, then the system has infinite solutions. It would be hard finding this answer when we use the Cramer's Rule so instead we use the Gauss Elimination. Considering the equations:

x + y = 3 and 2x + 2y = 6
Determinant of the equations are 
| 1 1 | 
| 2 2 | = 0

the numerator determinants would be
| 3 1 | . .| 1 3 | 
| 6 2 | = | 2 6 | = 0.
Executing Gauss Elimination, any two numbers, whose sum is 3, would satisfy the given system. For instance (3, 0), (2, 1) and (4, -1). Therefore, it would have infinitely many solutions.