Let $A$ be a matrix of order $2 \times 2$ such that $A^2 = O$. Then $A^2 - (a + d)A + (ad - bc)I$ is equal to
Step-by-Step Solution
Key Concept: Every $2 \times 2$ matrix satisfies the identity $A^2 - (a+d)A + (ad-bc)I = O$ regardless of any specific conditions like $A^2 = O$.
Let $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$ <br/> $\Rightarrow A^2 - (a + d)A + (ad - bc)I$ <br/> $= \begin{bmatrix} a & b \\ c & d \end{bmatrix} \begin{bmatrix} a & b \\ c & d \end{bmatrix} - (a + d) \begin{bmatrix} a & b \\ c & d \end{bmatrix} + (ad - bc) \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$ <br/> $= \begin{bmatrix} a^2 + bc & ab + bd \\ ac + cd & bc + d^2 \end{bmatrix} - \begin{bmatrix} a^2 + ad & ab + bd \\ ac + cd & ad + d^2 \end{bmatrix} + \begin{bmatrix} ad - bc & 0 \\ 0 & ad - bc \end{bmatrix} = O$
Correct Answer: B