GATE CS 1987 Q1-xxiii — Linear Algebra: Matrices, Determinants, System of Linear Equations

A square matrix $A$ of size $n \times n$ is defined to be singular if its determinant is zero ($\det(A) = 0$).

By properties of matrices, the following statements are equivalent for a square matrix $A$:

• $\det(A) = 0$ (the matrix is singular).
• The rank of $A$ is strictly less than $n$.
• The rows of $A$ are linearly dependent.
• The columns of $A$ are linearly dependent.

Let's evaluate the given options:

• Options A and B: If the rows (or columns) are linearly independent, the rank of the matrix is $n$, which means $\det(A) \neq 0$ and the matrix is non-singular.
• Option C: If the rows are linearly dependent, the rank of the matrix is less than $n$. This guarantees that the determinant is zero, making the matrix singular.

Therefore, a square matrix is singular whenever its rows are linearly dependent.