Nilpotent Matrix

Question

A square matrix A is nilpotent if some positive integer power of A equals the zero matrix; the determinant of any nilpotent matrix must equal which value?

RankGuru · Accepted Answer

A nilpotent matrix satisfies A^k = O for some positive integer k, a condition that pins down its determinant in JEE Advanced. Taking determinants of A^k = O gives det(A)^k = det(O) = 0. Since the only number whose power is zero is zero itself, det(A) = 0. Hence every nilpotent matrix is singular. Equivalently, all eigenvalues of a nilpotent matrix are zero, so their product, the determinant, is zero. Option 1 would make A invertible, contradicting A^k = O. Option -1 similarly implies invertibility. Option the order of the matrix has no basis. Hence the determinant is 0. Plausibility check: a strictly upper-triangular matrix is nilpotent and visibly has zeros on its diagonal, giving determinant zero, which matches the general conclusion that nilpotency forces a vanishing determinant. Elementary row operations form the basis of Gaussian elimination, and knowing precisely how each operation, row swaps, scaling, and adding multiples, affects the determinant lets a student simplify a matrix while tracking its determinant exactly. The identical-row consequence is a special case of the deeper principle that any linear dependence among rows forces the determinant, and hence invertibility, to collapse to zero.

Nilpotent Matrix

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