A system of n equations in n unknowns AX = B is analyzed by examining the coefficient determinant D = det(A).

Case 1: D != 0 (Non-singular coefficient matrix) Unique solution exists. Use Cramer's Rule: $x_i$ = $\frac{D_i}{D}$ where $D_i$ is obtained by replacing the i-th column of A with B.

Case 2: D = 0 (Singular coefficient matrix)

• If $D_1$ = $D_2$ = $D_3$ = 0: Infinitely many solutions. The system is consistent but underdetermined. Express solutions in terms of a free parameter.
• If any $D_i$ != 0: No solution. The system is inconsistent.

Homogeneous Systems (AX = O):

• D != 0: Only trivial solution X = O
• D = 0: Infinitely many non-trivial solutions exist
• Special case: If #unknowns > #equations, non-trivial solutions always exist

Rank Method (Rouche-Capelli Theorem):

• rank(A) = rank([A|B]) = n: unique solution
• rank(A) = rank([A|B]) < n: infinite solutions with (n - rank) free variables
• rank(A) < rank([A|B]): no solution

JEE Problem Types:

1. Find parameter values for which system has no/unique/infinite solutions
2. For homogeneous systems, find the ratio x:y:z using cross-multiplication
3. Determine geometric interpretation (planes meeting at a point, along a line, or forming a prism)

Common Exam Strategy: When D = 0, always check ALL $D_i$ values. Students often stop at D = 0 and immediately conclude "infinite solutions" without checking consistency.