A square matrix is singular whenever A. The rows are linearly independent B. The columns are linearly independent C. The rows are linearly dependent D. None of the above

GATE 1987 · Engineering Mathematics · Matrix · medium

Answer: The answer is C: the rows are linearly dependent.

1. Recall the definition of a singular matrix: A square n x n matrix is singular if and only if its determinant is zero, equivalently when its rank is less than n.
2. Evaluate each option: A: Linearly independent rows => full rank => non-singular. B: Linearly independent columns => same conclusion. C: Linearly dependent rows => rank < n => det = 0 => singular. D: Not applicable since C is correct.