Correct : b,d

The vector e₂ = (0,1,0,...,0) being in the null space means Me₂ = 0. Multiplying any matrix M by the standard basis vector e₂ simply extracts the second column of M. So Me₂ = 0 tells us directly that the second column of M is all zeros.
Option A — False
If the second column is all zeros, then det(M) = 0, not 1. A determinant of 1 would require M to be non-singular, but a zero column makes it singular. A is false.
Option B — True
The determinant of any matrix with a zero column is always 0. This follows from the multilinearity of the determinant — scaling any column by 0 scales the determinant by 0. Since the second column is identically zero, det(M) = 0 always. B is always true.
Option C — False
Rank cannot be pinned to exactly 1. A matrix with a zero second column could have rank anywhere from 0 (all columns zero) to n−1 (only the second column zero, all others linearly independent). The rank is not determined solely by having one zero column. C is not always true.
Option D — True
The null space of M is a subspace of ℝⁿ. Since it contains the non-zero vector e₂, it must contain all scalar multiples of e₂ — including 2e₂, 3e₂, −e₂ and infinitely more. In particular it always contains at least two non-zero vectors. D is always true.
Correct answer: B and D ✓