Singular Matrices

A square matrix is singular precisely when its determinant equals zero, the defining JEE Advanced condition for non-invertibility. To find the values of k making the given matrix singular, one sets its determinant to zero and solves the resulting polynomial in k. The determinant is a cubic in k whose real roots are the singular values; the conceptual step is recognizing that singularity is governed by the determinant, not the trace or rank statement directly. Option trace measures the diagonal sum and does not determine invertibility. Option sum of diagonal cofactors relates to other invariants but is not the singularity condition. Option rank is a consequence of singularity, but the operational test that one sets to zero is the determinant itself. Hence singularity is obtained by setting the determinant to zero. Plausibility check: a singular matrix maps some non-zero vector to zero, which is equivalent to a vanishing determinant, so equating the determinant to zero is the precise algebraic condition, consistent with the definition of singularity.