If M is a square matrix with a zero determinant, which of the following…
2008
If M is a square matrix with a zero determinant, which of the following assertion(s) is/are correct?
(S1) Each row of M can be represented as a linear combination of the other rows.
(S2) Each column of M can be represented as a linear combination of the other columns.
(S3) MX = 0 has a nontrivial solution.
(S4) M has an inverse.
- A.
S3 and S2
- B.
S1 and S4
- C.
S1 and S3
- D.
S1, S2 and S3
Attempted by 9 students.
Show answer & explanation
Correct answer: D
Since det(M) = 0, the square matrix M is singular and does not have full rank.
Because the rank is less than the order of the matrix, the row vectors are linearly dependent and the column vectors are also linearly dependent. Hence S1 and S2 are true.
Also, det(M) = 0 implies that the homogeneous system MX = 0 has nonzero solutions, so S3 is true.
S4 is false because a square matrix has an inverse only when its determinant is nonzero.
Therefore, the correct assertions are S1, S2 and S3.