Step-by-Step Solution
Key Concept: Use the Cayley-Hamilton theorem for a 2x2 matrix, $A^2 - \text{tr}(A)A + |A|I = O$, to establish the relationship between the matrix, its trace, and its determinant.
If $A = O, \text{tr}(A) = 0$. suppose $A \neq O$ and $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \Rightarrow |A| = 0$ and $A^2 - (a + d)A + (ad - bc)I = 0 \Rightarrow a + d = 0$
Correct Answer: B