GATE CS 2014 Set 2 Q4 — Linear Algebra: Matrices, Determinants, System of Linear Equations

The given matrix $A$ is defined as the product of a column vector and a row vector:

This is an outer product of two vectors, $A = u v^T$, where $u$ is a $3 \times 1$ column vector and $v^T$ is a $1 \times 3$ row vector. Thus, $A$ is a $3 \times 3$ matrix.

Since $A$ is formed by multiplying a single column vector by a row vector, every row of $A$ is a scalar multiple of the row vector $v^T$:

Because all rows are linearly dependent, the rank of $A$ is $1$. For any $n \times n$ matrix with $n \ge 2$, if its rank is strictly less than $n$, the matrix is singular, and its determinant is zero.

Therefore, the determinant of $A$ is 0.