When does a matrix obtained from a system of linear equations lead to a unique solution?

A matrix derived from a system of linear equations leads to a unique solution when its determinant is non-zero. This is a fundamental principle in linear algebra. When the determinant is non-zero, it indicates that the matrix is invertible, implying that the linear equations represented by the matrix are independent and intersect at a single point in the solution space.

In practical terms, a non-zero determinant signifies that there are no redundant equations and that the equations do not contradict each other, allowing for exactly one solution to exist. Conversely, if the determinant is zero, it suggests that the system of equations may either have no solutions or infinitely many solutions, reflecting either a dependency among the equations or inconsistent equations that do not intersect at a point.

The other options describe scenarios that do not allow for a unique solution: a singular matrix (which is what a zero determinant indicates) suggests dependency, a zero row introduces a contradiction or additional dependencies, and multiple rows of the same values indicate redundancy in the equations, neither of which leads to a unique solution. Thus, the condition of having a non-zero determinant guarantees that a unique solution exists.