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. S1 and S4 B. S1, S2 and S3 C. S1 and S3 D. S1, S2 and S4

GATE 2008 · Engineering Mathematics · Matrix · medium

Answer: S1, S2, and S3 are all correct. S4 is false. Answer: D (S1, S2 and S3).

1. Analyze S1 and S2 - Linear dependence of rows and columns: Since det(M) = 0, rank(M) < n. This means rows are linearly dependent (S1 TRUE) and columns are linearly dependent (S2 TRUE). Each row/column can be expressed as a linear combination of the others.
2. Analyze S3 - Nontrivial solution to MX = 0: Since rank(M) < n, the null space has dimension >= 1, meaning MX = 0 has nontrivial solutions (S3 TRUE).
3. Analyze S4 - Existence of inverse: Since det(M) = 0, M is singular and has NO inverse (S4 FALSE).