System Of Linear Equations

A linear system Ax = b has a unique solution when the coefficient determinant is non-zero, the central JEE Advanced criterion via Cramer's rule. The coefficient matrix has rows (1,1,1),(1,2,3),(1,4,9). Its determinant expands to 1(2·9 - 3·4) - 1(1·9 - 3·1) + 1(1·4 - 2·1) = 1(18 - 12) - 1(9 - 3) + 1(4 - 2) = 6 - 6 + 2 = 2, which is non-zero. A non-zero determinant guarantees the matrix is invertible, so the system has exactly one solution. Solving gives x = 1, y = 2, z = 3. Option inconsistent requires a non-zero determinant condition to fail with contradictory equations. Option infinitely many solutions requires a zero determinant with consistent equations. Option homogeneous trivial is wrong since the right side is non-zero. Hence the system is consistent with a unique solution. Plausibility check: substituting x = 1, y = 2, z = 3 satisfies all three equations (6, 14, 36), confirming the unique solution implied by the non-zero determinant. Cramer's rule provides an explicit determinant formula for each unknown and is most efficient for small systems, while also exposing exactly when a unique solution exists through the non-vanishing of the coefficient determinant. For larger systems the same determinant condition governs solvability even though row reduction becomes the practical method, so the conceptual criterion remains determinant-driven throughout linear algebra.