Let A = \begin{pmatrix} m & n \\ p & q \end{pmatrix}, d = |A| \neq 0, |A - d(\text{adj } A)| = 0. Then
Step-by-Step Solution
Key Concept: Use the property of the adjoint of a 2x2 matrix: adj(A) = [[q, -n], [-p, m]] and the determinant condition |A - d(adj A)| = 0.
Given A = [[m, n], [p, q]], d = mq - np. adj(A) = [[q, -n], [-p, m]]. A - d(adj A) = [[m - dq, n + dn], [p + dp, q - dm]]. The determinant is (m - dq)(q - dm) - (n + dn)(p + dp) = 0. Expanding this leads to the condition 1 + d^2 = m^2 + q^2.
Correct Answer: 4