Determinant Of A Product

The determinant of a product equals the product of determinants, det(AB) = det(A)·det(B), a multiplicative property fundamental to JEE Advanced matrix algebra. With det(A) = 3 and det(B) = -2, we compute det(AB) = 3 · (-2) = -6. This property holds for square matrices of the same order and reflects how composing linear maps multiplies their volume-scaling factors. Option 6 drops the negative sign of det(B). Option 1 wrongly treats the determinant as additive in a normalized way. Option -5 incorrectly adds the determinants as 3 + (-2). Hence det(AB) = -6. Plausibility check: since the determinant measures how a linear map scales volume, applying B then A scales by (-2) then by 3, multiplying to -6, and the sign indicates an orientation reversal, consistent with the negative det(B). Singularity, zero determinant, linear dependence of rows, and the existence of a non-trivial null vector are four equivalent descriptions of the same phenomenon, and recognizing their interchangeability is central to JEE Advanced matrix theory. The operational test of setting the determinant to zero is favoured because it converts an abstract invertibility question into a concrete polynomial equation in the unknown parameter.