1. The problem asks to identify the condition under which a square matrix $A$ is singular. 2. A matrix is singular if it does not have an inverse. 3. The determinant of a matrix, denoted $|A|$, is a scalar value that can be used to determine if the matrix is invertible. 4. The key rule is: A matrix $A$ is singular if and only if its determinant $|A| = 0$. 5. Therefore, the correct answer is option C: $|A| = 0$.