Key Concept: Conditions for Solutions of a System of Linear Equations (Cramer's Rule)

For a system of linear equations in matrix form $AX = B$, where $A$ is the coefficient matrix, $X$ is the variable matrix, and $B$ is the constant matrix, we use determinants to determine the nature of solutions. Let $\Delta = \det(A)$ be the determinant of the coefficient matrix. Let $\Delta_x, \Delta_y, \Delta_z$ (or $\Delta_1, \Delta_2, \Delta_3$ as sometimes denoted) be the determinants obtained by replacing the column of coefficients for $x, y, z$ respectively with the constant terms from matrix $B$.

The conditions are:

1. **Unique