Let $A = \begin{pmatrix} m & n \\ p & q \end{pmatrix}$, $d = |A| \neq 0$ and $|A - d(\text{Adj}A)| = 0$. Then
Step-by-Step Solution
Key Concept: Compute $A - d(\text{Adj}A)$ explicitly and set its determinant to zero
For $2\times 2$: $\text{Adj}A = \begin{pmatrix}q&-n\\-p&m\end{pmatrix}$. $A - d\,\text{Adj}A = \begin{pmatrix}m-qd & n(1+d)\\p(1+d) & q-md\end{pmatrix}$. Setting determinant to 0: $(m-qd)(q-md) - np(1+d)^2 = 0$. Expanding and using $d = mq-np$: after simplification $(1+d)^2 = (m+q)^2$. Answer: (1)
Correct Answer: $(1+d)^2 = (m+q)^2$