Determinant Using Row Operations

Elementary row operations have predictable effects on determinants, and a matrix with two identical rows has determinant zero, a property heavily used in JEE Advanced simplifications. Adding a multiple of one row to another does not change the determinant, so if such operations make two rows identical, the determinant of the resulting matrix equals that of the original. A determinant with two identical rows is zero because swapping those rows both negates the determinant and leaves it unchanged, forcing it to equal its own negative. Option 1 ignores the identical-row property. Option the product of the diagonal applies only to triangular matrices. Option undefined is wrong since determinants are always defined for square matrices. Hence the determinant is 0. Plausibility check: two identical rows mean the rows are linearly dependent, so the matrix is singular, and singular matrices have determinant zero, consistent with the row-operation reasoning. Nilpotent matrices have an entirely zero eigenvalue spectrum, which is why every nilpotent matrix is singular and has zero trace as well as zero determinant. They model genuinely degenerate transformations that eventually annihilate every vector, and their canonical strictly triangular form makes the vanishing determinant visible at a glance, reinforcing the algebraic argument from the defining power relation.