Let π > 1. Consider an πΓπ matrix π with its elements from β. Let theβ¦
2026
Let π > 1. Consider an πΓπ matrix π with its elements from β. Let the vector (0,1,0,0,β¦,0)ββπ be in the null space of π.
Which of the following options is/are always correct?
- A.
Determinant of π is 1
- B.
Determinant of π is 0
- C.
Rank of π is 1
- D.
There are at least two non-zero vectors in the null space of π
Attempted by 7 students.
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Correct answer: B, D
The vector (0, 1, 0, ..., 0) is the second standard basis vector e_2. Since e_2 is in the null space of M, we have M e_2 = 0. But M e_2 is exactly the second column of M, so the second column of M is the zero column.
A matrix with a zero column is singular, hence det(M) = 0. Also, because e_2 is in the null space, every scalar multiple c e_2 is also in the null space. Over R, this gives infinitely many non-zero vectors in the null space, so at least two non-zero vectors certainly exist.
The rank need not be 1; it could be as large as n - 1 if the remaining columns are independent. Therefore the always-correct options are 2 and 4.