Homogeneous System Non-trivial Solution

A homogeneous system Ax = 0 has a non-trivial solution exactly when the coefficient determinant is zero, a key JEE Advanced linear-algebra criterion. A homogeneous system always admits the trivial solution x = 0; additional non-zero solutions exist only when the columns are linearly dependent, equivalently when det(A) = 0. For the given matrix with parameter k, one finds the values of k that make the determinant vanish, and those values permit non-trivial solutions. Option trace equals 0 is unrelated to the existence of non-trivial solutions. Option k equals 0 always is a specific guess not derived from any condition. Option the system never has non-trivial solutions is false, since a vanishing determinant guarantees them. Hence the condition is that the determinant of the coefficient matrix equals zero. Plausibility check: when det(A) = 0 the rows are linearly dependent, so one equation is redundant, leaving a free variable that generates infinitely many non-trivial solutions, consistent with the criterion. Homogeneous systems are governed entirely by the coefficient determinant, with a zero determinant signalling linear dependence among the equations and hence a continuum of non-trivial solutions. This determinant condition is the workhorse for eigenvalue problems too, since demanding non-trivial solutions of A minus lambda I times x equal to zero is exactly what produces the characteristic equation studied later.